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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system At first, we study the resolution procedure by blowing-up an \(n\)- dimensional linear system of two variable power series. The main application of this study is an algebraic characterization of the linear system geometry by means of Zariski's complete ideal theory.
In the second part, we study more specifically the desingularization of the one-dimensional linear systems (pencils). We explain the parallelism between the geometry of a pencil and the associated meromorphic foliation. Finally, we characterize the set of pencils associated to a given foliation with a non-constant first integral. We work on the formal power series ring over the complex field but all the results are true on a two-dimensional, local, regular, integrally closed ring over an algebraically closed field. blowing-up an \(n\)-dimensional linear system; resolution; meromorphic foliation; formal power series ring Global theory and resolution of singularities (algebro-geometric aspects), Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\), Formal power series rings, Divisors, linear systems, invertible sheaves Linear systems of power series, pencils and associated foliations | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The article originated from a course which the author gave at a winter school in Barcelona in 2006. Following the same plan of exposition of the material as in an earlier joint article with \textit{A. Bravo} and \textit{S. Encinas} [Rev. Mat. Iberoam. 21, No. 2, 349--458 (2005; Zbl 1086.14012)] (but this time in the style of a sequence of lectures and extended in content), the author thoroughly treats constructive desingularization in characteristic zero starting from the statement of the main results and the underlying basic definitions and techniques. He then proceeds to the description of the complete resolution process based on the notion of resolution invariants resp. a resolution function, whose structure reflects the inductive nature of desingularization (by descent in dimension of the ambient space) and is explained on the basis of its building blocks, suitable upper semicontinuous functions.
While the focus of the already described part is on the local construction of the resolution process and invariants, the last part concentrates on globalizing those local results. Key notions in this context are Rees algebras and differential algebras, which are introduced and subsequently used to reformulate the objects of study of the previous part in the global context. In addition to a proof following the approach outlined in Kollàr's lecture notes on resolution of singularities, the author also presents -- for the so-called `simple case' -- a different approach, which is not restricted to the case of characteristic zero and might contribute to the still unsolved problem of desingularization in positive characteristic. resolution of singularities; desingularization; resolution function; resolution invariant; Rees Algebra; differential algebra; constructive resolution Villamayor, O., An introduction to constructive desingularization (Notes) Global theory and resolution of singularities (algebro-geometric aspects), Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics Notes on constructive desingularization | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A local system \(L\) on a nonempty Zariski open subset \(U\) of \(\mathbb{P}^{1,\mathrm{an}}_{\mathbb{C}}\) that is determined by its local monodromy is called a physically rigid local system. \textit{N. M. Katz} [Rigid local systems. Princeton, NJ: Princeton Univ. Press (1996; Zbl 0864.14013)] proved that if \(L\) is irreducible, then being physically rigid is equivalent to the vanishing of the intersection (= parabolic) cohomology \(H^1_p(\mathrm{End}(L)):=H^{1}(\mathbb{P}^{1,\mathrm{an}}_{\mathbb{C}}, j_{\ast}\mathrm{End}(L))\), where \(j\colon U \to \mathbb{P}^{1}_{\mathbb{C}}\) is the open immersion. In a precise sense, the intersection cohomology space can be regarded as the tangent space to \(L\) in a suitable moduli stack.
The analogue of a local system in rigid cohomology is an overconvergent isocrystal. It is also reasonable to regard the restriction of the isocrystal to the Robba rings of the punctures as a replacement of data of local monodromy. Thus one can define the notion of a physically rigid isocrystal. For an absolutely irreducible overconvergent F-isocrystal, \textit{R. Crew} [Doc. Math. 22, 287--296 (2017; Zbl 1391.14038)] showed that ``vanishing parabolic cohomology'' \(\Longrightarrow\) ``physically rigid'', i.e., the analogue of Katz's theorem holds.
In the article under review, the author studies the local deformation theory of isocrystals. The author proves that the rigid cohomology groups \(H^1(\mathrm{End}(L))\), \(H^1_c(\mathrm{End}(L))\), \(H^1_{p}(\mathrm{End}(L))\) are Zariski tangent spaces of some functors of artin rings, and these deformation functors have hulls if, for example, \(\mathrm{End}(L)\) admits a Frobenius structure. If moreover \(L\) is absolutely irreducible, some smoothness and pro-representability results about these functors are proved. \(p\)-adic cohomology; isocrystals; arithmetic \(D\)-modules; deformation theory; Hochschild cochain complex \(p\)-adic cohomology, crystalline cohomology, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, \(p\)-adic differential equations, Local ground fields in algebraic geometry, Rigid analytic geometry Deformations of overconvergent isocrystals on the projective line | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(R\subset S\) be two regular local rings of dimension \(d\geq 2\) having the same field of fractions. Suppose that S dominates R and if \(d>2\) then R is assumed excellent. Then for every strictly decreasing chain \(R\subset S_ m\subset...\subset S_ 1\subset S\) such that
(1) all \(S_ i\) are regular local rings of dimension d,
(2) \(S_{i+1}\) dominates \(S_ i\) for all i, \(0\leq i\leq m\), where \(S_{m+1}:=R\) and \(S_ 0:=S,\)
the integer m is bounded by a positive integer N depending just on R, S. This bound is explicitly computed for some extensions and nice examples are given.
The above result extends the author's previous one [see \(corollary\quad (4,10)\) from Trans. Am. Math. Soc. 299, 513-524 (1987; Zbl 0659.13006)] which says that any factorization of a local birational morphism \(f:\quad Spec(S)\to Spec(R)\) of nonsingular (affine) schemes of arbitrary dimension via other nonsingular schemes must be finite in length (in the surface case the classical local factorization theorem of Zariski and Abhyankar says that such factorization is unique). monoidal transformation; excellent ring; regular local rings; factorization of a local birational morphism Johnston, B.: The uniform bound problem for local birational nonsingular morphisms. Trans. amer. Math. soc. 312, No. No.1, 421-431 (1989) Local structure of morphisms in algebraic geometry: étale, flat, etc., Regular local rings, Rational and birational maps, Commutative ring extensions and related topics The uniform bound problem for local birational nonsingular morphisms | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system This article deals with the exposition of a new approach to the problem of choice of suitable centers in a Hironaka style desingularization, i.e. in a resolution of singularities by a sequence of blowing ups. Although constructive proofs of resolution of singularities in characteristic zero have been known and refined for some time now, there are major obstructions to generalizing them to the open case of positive characteristic, e.g. the key inductive argument on the descent in ambient dimension involves `hypersurfaces of maximal contact' which do not exist in positive characteristic. The approach explained in this article replaces the restriction to such hypersurfaces by local projections which exist in any characteristic and which also allow related resolution invariants with significantly easier globalization. It is shown that in characteristic zero this construction provides the same information as the traditional one. A drawback and still open problem of this approach is that it only simplifies the singularities to a monomial case; this in turn is not as straightforward as in Hironakas classical approach and its resolution is not shown in all generality, but only under some additional conditions. As a sample application the case of 2-dimensional embedded schemes is treated. constructive resolution of singularities; desingularization; choice of center; projections ,\textit{Techniques for the study of singularities with applications to resolution of 2-dimensional} \textit{schemes}, Math. Ann. 353 (2012), no. 3, 1037--1068. http://dx.doi.org/10.1007/s00208-011-0709-5.MR2923956 Global theory and resolution of singularities (algebro-geometric aspects) Techniques for the study of singularities with applications to resolution of 2-dimensional schemes | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \({\mathcal A}=\{H_1,\ldots H_n\}\) be an arrangement of complex hyperplanes, with complement \(M=M({\mathcal A})\). Let \(\mathcal L\) be a rank-one complex local system over \(M\). The local system \(\mathcal L\) is determined by its monodromy \(H^1(M) \to {\mathbb C}^*\), hence by a point \(t=(t_1,\ldots, t_n)\in ({\mathbb C}^*)^n.\)
This paper concerns the effect of isotopies of the arrangement \({\mathcal A}\) on the local system cohomology \(H^*(M,{\mathcal L}_t),\) in particular its dependence on \(t\).
Let \( B\) be a smooth component of the realization space of a simple matroid \(G\) on \(n\) points. Points \(b \in B\) give rise to arrangements of \(n\) labelled hyperplanes with the same underlying matroid \(G\), and with diffeomorphic complements. Fix \(t\in ({\mathbb C}^*)^n\) and let \((\lambda_1,\ldots,\lambda_n)\in {\mathbb C}^n\) with \(t_j=\text{exp}(-2\pi i \lambda_j)\). One obtains for each \(q\) a bundle of vector spaces over \( B\), whose fiber over \({\mathcal A}_b\) is the local system cohomology \(H^q(M({\mathcal A}_b),{\mathcal L}_t)\). The authors compute the monodromy of this bundle, and its infinitesimal generator, the Gauss-Manin connection. The main result is that the eigenvalues of the Gauss-Manin connection are integral linear combinations of the weights \(\lambda_j,\) confirming a conjecture of H.~Terao. The classical example arises from the family of discriminantal (Selberg-type) arrangements, parametrized by the configuration space \(b_k\) of \(k\) labelled points in the plane. In this case the Gauss-Manin connection was described by \textit{K. Aomoto} [J. Math. Soc. Japan 39, 191--208 (1987; Zbl 0619.32010)]. (N.B., In general, the realization space of \(G\) need not have any smooth components.)
Let \(\Lambda={\mathbb C}[x_1^{\pm 1}, \ldots, x_n^{\pm 1}]\) denote the ring of Laurent polynomials. In previous work Cohen and Orlik used stratified Morse theory to construct a combinatorial cochain complex \((K^\bullet({\mathcal A}),\Delta(x))\) of \(\Lambda\)-modules and \(\Lambda\)-homomorphisms, such that specialization \({ x_j} \mapsto { t_j}\) yields a complex whose cohomology is \(H^*(M,{\mathcal L}_t).\) Here the authors develop a parametrized version of this construction. The universal \(\Lambda\)-complexes \((K^\bullet({\mathcal A}_b),\Delta(x))\) form a bundle over \( B\), with a universal monodromy operator \(\Phi^\bullet(x)\) acting by chain automorphisms on \((K^\bullet({\mathcal A}_{ b}),\Delta(x))\). The linearization at \(t={ 1}\) of \((K^\bullet({\mathcal A}_b),\Delta(x))\) is the Aomoto complex of \({\mathcal A}_b\), with ground ring \({\mathbb C}[y_1,\ldots, y_n].\) The linearization of \(\Phi^\bullet(x)\) acts on the Aomoto complex, inducing the Gauss-Manin connection upon specialization \(y_j\mapsto \lambda_j.\) Fixing \(\gamma\in \pi_1(b)\), an eigenvalue \(r(x)\) of \(\Phi^q(x)(\gamma)\) determines a nowhere zero analytic function \(({\mathbb C}^*)^n\to {\mathbb C}\) and thus is given by \(r(x)=x_1^{m_1}\cdots x_n^{m_n}\) for some \((m_1,\ldots, m_n)\in {\mathbb Z}^n.\) It follows that the eigenvalues of the Gauss-Manin connection are integral linear combinations of the weights \(\lambda_i\).
The constructions are illustrated with an extended example. Gauss-Manin connection; hyperplane arrangement; local system Cohen D, Orlik P. Gauss-Manin connections for arrangements, I. Eigenvalues. Compositio Math, 136:299--316 (2003) Relations with arrangements of hyperplanes, Structure of families (Picard-Lefschetz, monodromy, etc.), Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Homology with local coefficients, equivariant cohomology Gauss-Manin connections for arrangements. I: Eigenvalues | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system [For the entire collection see Zbl 0721.00009.]
These notes represent a survey of the main points of the authors' explicit proof of canonical desingularization of an algebraic subvariety (resp. analytic subspace) \(X\) of an algebraic (resp. analytic) manifold \(M\), in characteristic zero. The full details of this result are planned for a forthcoming work `Canonical desingularization in characteristic zero: a simple constructive proof'. The authors' result is essentially a new proof of Hironaka's theorem, although they also give an explicit resolution algorithm. The centres of the blowings-up used in the desingularization are determined by a local invariant of the singularity of \(X\), defined over a sequence of blowings-up.
The first section of this paper describes the general strategy of the proof and ends with a precise statement of the main theorem. --- The second section recalls the definitions and basic notions involved in resolution of singularities (analytic space, blowing-up, strict transform, normal crossings, etc.). --- The third section introduces the local invariant of a singularity of \(X\) and begins an analysis of this invariant. --- The fourth section gives a key part of the proof that it is in fact invariant. Much of these notes deal with the special case where \(X\) is a hypersurface, and the general case involves a reduction to this case. canonical desingularization of an algebraic subvariety; resolution algorithm Zhou, X. Y., Zhu, L. F.: An optimal \(L\)\^{}\{2\} extension theorem on weakly pseudoconvex Kähler manifolds. To appear in J. Differential Geom. Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Effectivity, complexity and computational aspects of algebraic geometry A simple constructive proof of canonical resolution of singularities | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The author considers linear systems \(\Lambda\) of hypersurfaces on a projective space \(\mathbb{P}^r(\mathbb{C})\) and the associated rational map \(\Phi:=\Phi_\Lambda :\mathbb{P}^r\to\mathbb{P}^\alpha .\) In Theorem 1 it is shown that the existence of linear syzygies for the homogeneuos ideal of the base locus of \(\Lambda\) has strong consequences for the fibers of \(\Phi.\) The result reminds a well-known fact according to which the \(K_d\) property of the base locus implies the linearity of the fibers of \(\Phi.\) The next main result, Theorem 3, concerns the special case of linear systems of quadrics. It states that, when the Gauss map of \(\Phi(\mathbb{P}^r)\) is degenerate, the inverse image of its non-trivial fibers is a linear space in \(\mathbb{P}^r.\) A number of interesting applications and examples are given. quadrics; linear syzygies; rational map. A. Alzati, Special linear systems and syzygies, Collect. Math., 59:239-254, 2008. Rational and birational maps, Determinantal varieties Special linear systems and syzygies | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(R\subset S\) be local domains essentially of finite type over a field \(k\) with char \(k=0\) and \(V\) a valuation ring of the quotient field \(K\) of \(S\). Then there exist sequences of monomial transforms \(R\rightarrow R'\), \(S\rightarrow S'\) along \(V\) such that \(R'\), \(S'\) are regular local rings, \(S'\) dominates \(R'\) and there exist regular system of parameters \((y_1,\ldots,y_n)\) in \(S'\), \((x_1,\ldots,x_m)\) in \(R'\), units \(\beta_1,\ldots,\beta_n\in S'\) and a \(m\times n\) matrix \((c_{ij})\) of non-negative integers such that Rank\((c_{ij})=m\) and \(x_i=\Pi_{j=1}^ny_j^{c_{ij}}\beta_i\), \(1\leq i\leq m\).
This is the most general possible relative ``Local Uniformization Theorem for mappings'', the case \(R=k\) being given by Zariski. When \(K\) is a finite extension of the quotient field of \(R\) the result was already stated by the author in [``Local monomialization and factorization of morphisms'', Astérisque 260 (1999; Zbl 0941.14001)]. The above result is used to the construction of a monomialization by quasi-complete varieties, which proves a local version of the toroidalization conjecture of \textit{D. Abramovich, K. Karu, K. Matsuki} and \textit{J. Wlodarczyk} [J. Am. Math. Soc. 15, 531--572 (2002; Zbl 1032.14003)]. resolution of singularities; local uniformization theorem; toroidalization Cutkosky, SD, Local monomialization of transcendental extensions, Ann. Inst. Fourier, 55, 1517-1586, (2005) Global theory and resolution of singularities (algebro-geometric aspects), Valuations and their generalizations for commutative rings, Singularities in algebraic geometry, Rational and birational maps Local monomialization of transcendental extensions. | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(X \subset \mathbb{P}^N\) be a smooth nondegenerate projective variety of dimension \(n\) (\(\geq 2\)), codimension \(e\) and degree \(d\) (over an algebraically closed field of characteristic zero). It is known that the linear system \(|{\mathcal O}_X(d-n-2)\otimes \omega_X^\vee|\) is base point free, as their elements can be seen as the double point divisors of linear projections onto hypersurfaces of \(\mathbb{P}^{n+1}\). Moreover, it has been proved (see the Introduction of the paper under review and references therein) that they separate points unless \(X \subset \mathbb{P}^N\) is of a particular type (a projection of a so called \textit{Roth variety}). The main purpose of this paper is to prove further positivity results on these linear systems. To be precise: It is shown that the base locus of \(|{\mathcal O}_X(d-n-e-1)\otimes \omega_X^\vee|\) is a finite set except if \(X \subset \mathbb{P}^N\) belongs to a finite list of projective varieties (completely stated, see Thm. 4). For the study of these exceptions, varieties whose generic inner projections have exceptional divisors are classified.
Some applications of these results are also provided: inequalities for the delta and sectional genera, property \((N_{k-d+e})\) for \({\mathcal O}_X(k)\) and new evidences of the regularity conjecture, in fact that \({\mathcal O}_X\) is \((d-e)\)-regular. double point divisors; base locus; linear projections; regularity Noma, A., Generic inner projections of projective varieties and an application to the positivity of double point divisors, Trans. Amer. Math. Soc., 336, 4603-4623, (2014) Classical problems, Schubert calculus, Projective techniques in algebraic geometry Generic inner projections of projective varieties and an application to the positivity of double point divisors | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The main point is a generalization of the author's previous paper [ibid. 1-74 (1990; see the preceding review)] to the non-hypersurface case. For a given ideal \({\mathfrak a}\) of a complete regular local ring of characteristic zero there is a polyhedral filtration satisfying a certain minimality property. It gives a coordinate-free definition of some Newton polyhedron related to \({\mathfrak a}\). The construction is based on Hironaka's notion of a standard base, see \textit{H. Hironaka}, Ann. Math., II. Ser. 79, 109-326 (1964; Zbl 0122.386). For application related to the theory of maximal contact [in the sense of \textit{J. Giraud}, see Math. Z. 137, 285-310 (1974; Zbl 0275.32003) see the author's reprint [Canonical formal uniformization in characteristic zero (Hebrew Univ. Jerusalem)]. polyhedral filtration; Newton polyhedron; standard base; maximal contact Boris Youssin, Newton polyhedra of ideals, Mem. Amer. Math. Soc. 87 (1990), no. 433, i -- vi, 75 -- 99. Toric varieties, Newton polyhedra, Okounkov bodies, Ideals and multiplicative ideal theory in commutative rings, Relevant commutative algebra Newton polyhedra of ideals | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Recall that a linear system of hypersurfaces of some degree \(d\) in the projective space \(\mathbb{P}^n\) is called special if its dimension is larger than its expected dimension. For special linear systems \({\mathfrak L}(d, m_1,\dots,m_r)\) of hypersurfaces of degree \(d\) passing through a scheme \(Z=m_1p_1+\cdots+m_rp_r\) of fat points in general position in \(\mathbb{P}^2\), there exists a well-studied and partially proven characterization due to B. Harbourne and A. Hirschowitz [see \textit{B. Harbourne}, Can. Math. Soc. Conf. Proc. 6, 95--111 (1986; Zbl 0611.14002) and \textit{A. Hirschowitz}, J. Reine Angew. Math. 397, 208--213 (1989; Zbl 0686.14013)]. The authors study the analogous linear systems in \(\mathbb{P}^3\). Their main tool is the cubic Cremona transformation Cr :\((x_0:x_1: x_2:x_3)\mapsto(x_0^{-1}:x_1^{-1};x_1^{-1}:x_3^{-1})\). They describe the action of Cr on the Picard group of the blow-up \(X\) of \(\mathbb{P}^3\) at \(\{p_1,\dots,p_r\}\) and use it to bring the linear systems into a standard form. For linear systems in standard form, they present a conjectural characterization of the special ones. In [\textit{C. De Volder} and \textit{A. Laface}, J. Algebra 310, No. 1, 207--217 (2007; Zbl 1113.14036)], this conjecture has been verified for \(r\leq 8\) fat points. The authors also apply their conjecture to the ``homogeneous case'' \({\mathfrak L}(d,m,\dots,m)\) and provide further evidence for it in various other cases. Cremona transformation; virtual dimension; fat point scheme; linear systems Laface, A.; Ugaglia, L., On a class of special linear systems of \(\mathbb{P}^3\), Trans. Amer. Math. Soc., 358, 5485-5500, (2006), (electronic) Divisors, linear systems, invertible sheaves On a class of special linear systems of \(\mathbb{P}^3\) | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Over fields of characteristic zero, resolution of singularities is achieved by means of an inductive argument, which is sustained on the existence of the so called hypersurfaces of maximal contact. We report here on an alternative approach which replaces hypersurfaces of maximal contact by generic projections. Projections can be defined in arbitrary characteristic, and this approach has led to new invariants when applied to the open problem of resolution of singularities over arbitrary fields. We show here how projections lead to a form of elimination of one variable using invariants that, to some extent, generalize the notion of discriminant.
This exposition draws special attention on this form of elimination, on its motivation, and its use as an alternative approach to inductive arguments in resolution of singularities. Using techniques of projections and elimination one can also recover some well known results. We illustrate this by showing that the Hilbert-Samuel stratum of a d-dimensional non-smooth variety can be described with equations involving at most d variables.
In addition this alternative approach, when applied over fields of characteristic zero, provides a conceptual simplification of the theorem of resolution of singularities as it trivializes the globalization of local invariants. resolution of singularities; Rees algebras Bravo, A; Villamayor U, OE, Elimination algebras and inductive arguments in resolution of singularities, Asian J. Math., 15, 321-355, (2011) Global theory and resolution of singularities (algebro-geometric aspects), Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics Elimination algebras and inductive arguments in resolution of singularities | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A new proof of Hironaka's theorem on resolution of singularities is given. There are already several different constructive approaches [cf. \textit{E. Bierstone} and \textit{P. Milman}, Invent. Math. 128, 207--302 (1997; Zbl 0896.14006) or \textit{O. Villamayor}, Ann. Sci. Éc. Norm. Supér., IV. Sér 22, 1--32 (1989; Zbl 0675.14003)]. The resolution process is based on the choice of an invariant which measures the singularities and drops under blowing up the maximal stratum of this invariant. The choice of the invariant and the way to compute it makes the difference between the approaches to resolve singularities. The definition of the invariant is quite involved. It is defined inductively using the knowledge of the resolution process up to this moment. The induction defining the invariant is given by the intersection of the variety with a so-called hypersurface of maximal contact. The choice of this hypersurface is not canonical and some effort is needed to define the invariant in a canonical way.
The approach of this paper is similar to the approach of Bierstone and Milman but based additionally on two observations. The resolution process defined as a sequence of blowing-ups the ambient spaces can be applied simultaneously to a class of equivalent singularities obtained by simple modifications, i.e. to resolve a singularity it is allowed to tune it before starting: In the equivalence class a convenient representative given by a so-called homogenized ideal is chosen. The restrictions of homogenized ideals to different hypersurfaces of maximal contact define locally analytically isomorphic singularities. resolution of singularities; algorithmic resolution Włodarczyk, Jarosław, Simple Hironaka resolution in characteristic zero, J. Amer. Math. Soc., 18, 4, 779-822, (2005) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Local complex singularities, Modifications; resolution of singularities (complex-analytic aspects) Simple Hironaka resolution in characteristic zero | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Linear systems \({\mathcal L} = {\mathcal L}_{n,d}(m_1,\dots,m_s)\) of hypersurfaces of degree \(d\) in \({\mathbb P}^n\) with assigned multiple points \(P_1,\dots, P_s\) of multiplicities \(m_1,\dots,m_s\), are studied. For such a system, its expected dimension is edim \({\mathcal L} = \)max\(\{{n+d\choose n} - \sum ^s_{i=1}{n+m_i -1 \choose n}, -1\}\); when its actual dimension is greater, we say that the linear system is special. Let \(Z \subset {\mathbb P}^n\) be the scheme (of ``fat points'') associated to \({\mathcal L} \); we will use \({\mathcal L} \), by abuse of notation, also for the ideal sheaf \({\mathcal I}_Z \otimes {\mathcal O}_{{\mathbb P}^n}(d)\); to say that the linear system is special is equivalent to the fact that \(h^0({\mathcal L},{\mathbb P}^n)h^1({\mathcal L},{\mathbb P}^n) > 0\).
In the paper, the definition of a new expected dimension of a linear system is given, namely its expected linear dimension, ldim \({\mathcal L}\) , which takes into account the contribution of the linear base locus, and thus the notion of linear speciality is introduced and studied. We always have dim \({\mathcal L}\geq\) ldim \({\mathcal L} \geq \) edim \({\mathcal L}\). Sufficient conditions for a linear system to be linearly non-special for an arbitrary number of points and necessary conditions for a small number of points are given; the main tool for those results is the study of blow-ups of \({\mathbb P}^n\) at linear subspaces which are in the base locus of the system. linear systems; fat points; base locus; linear speciality; effective cone Brambilla, MC; Dumitrescu, O; Postinghel, E, On a notion of speciality of linear systems in \(\mathbb{P}^n\), Trans. Am. Math. Soc., 367, 5447-5473, (2015) Divisors, linear systems, invertible sheaves, Hypersurfaces and algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry On a notion of speciality of linear systems in \(\mathbb{P}^{n}\) | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The aim of this paper is to construct a minimal injective resolution for an Artin-Schelter regular algebra of dimension 2, that is, for a twisted, homogeneous coordinate ring of the projective line: \(B(X, \sigma, L)\), where \(X = \mathbb{P}^1_k\), \(\sigma\) is an automorphism of \(X\), \(L={\mathcal O}_X (1)\). The first step is a unique factorization result for \(B(X, \sigma, L)\), generalizing the classical unique factorization in \(A = B(X,i,L)\), where \(i\) is the identity: The trick consists in substituting irreducible elements with products of irreducible elements in the same orbit. The classical minimal projective resolution:
\[
0 \to A \to K_A @>\alpha>>\bigoplus_{p \in \mathbb{P}^1_k} (K_A/A_p) \to A'(2) \to 0
\]
is now replaced by:
\[
0 \to B \to K_B @>\alpha>> \bigoplus_{\text{orbits} w} (K_B/B_w) \to B'(2) \to 0
\]
where: \(K_B\) is the graded quotient ring of \(B\), \(B' = \Hom_k (B,k)\), \(B_w\) is the localization of \(B\) at an orbit \(w\) of points of \(\mathbb{P}^1\) under the action of \(w\). The paper ends with an appendix on Serre duality for regular algebras. minimal injective resolution; twisted homogeneous coordinate ring of the projective line; Serre duality; regular algebras Ajitabh, K.: Residue complex for regular algebras of dimension 2. J. algebra 179, 241-260 (1996) Complexes, Relevant commutative algebra Residue complex for regular algebras of dimension 2 | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system This is an expository article, where the author presents an informal introduction to the theory of resolution of singularities of algebraic varieties (in characteristic zero) and the closely related question of principalization of ideals. The approach is constructive, or algorithmic.
As is common in this type of work, one considers instead a similar problem for more technical auxiliary objects. In the present article, these are the so-called singular mobiles. These are systems \((W, \mathcal I,c,D,E)\), where \(W\) is a regular variety, \(\mathcal I\) a coherent sheaf of ideals on \(W\), \(c\) a positive integer and \(D=D_n, \dots, D_1\), \(E=E_n, \dots, E_1\) strings of normal crossing divisors satisfying certain conditions. These mobiles are useful to formalize a critical inductive step in the resolution process. A full presentation of this theory is given in \textit{S. Encinas} and \textit{H. Hauser} [Comment. Math. Helv. 77, 821--845 (2002; Zbl 1059.14022)].
The article under review includes introductory examples, a discussion of the basic concepts, blowing-ups, transforms, order and other invariants, and the notion of mobiles, as well as many examples. A large bibliography on the subject follows. This paper, together with \textit{H. Hauser} [Bull. Am. Math. Soc. 40, 323--403 (2003; Zbl 1030.14007)], is a good introduction to the subject. singularity; resolution; blowing-up; mobile Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Desingularization of ideals and varieties | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A field \(k\) (of positive characteristic \(p\)) is \textit{differentiably finite over a perfect subfield \(k_0\)}, or a DF-field, if \(\Omega ^1 _{k/{k_0}}\) is a finite dimensional \(k\)-vector space. For instance, a function field over a perfect field is a DF-field.
The authors recently proved resolution of singularities for a quasi-projective algebraic \(X\) variety of dimension three (a 3-fold) over a DF-field \(k\) (in the sense that there is a projective morphism \(\pi : {\tilde X} \to X\), inducing an isomorphism off \(S\), the singular locus of \(X\), such that \(\pi ^{-1}(S)\) is a divisor in \({\tilde X}\) with strict normal crossings). This theorem improves previously known results on resolution of 3-folds in characteristic \(p\), which required stronger assumptions on the base field.
In their proof the authors use, following a strategy proposed by Zariski, and successfully followed by himself in characteristic zero, valuation-theoretic methods: prove local uniformization first and then by patching deduce the theorem. This proof is presented in two parts. In the present paper they show how, for a 3-fold over any field of characteristic \(p > 0\), desingularization follows from a special uniformization result. Namely, essentially it says that if one can prove local uniformization for valuations \(W \) which dominate the local ring of an Artin-Schreier (A-S) or a purely inseparable (p.i.) singularity, then desingularization is available. An A-S (resp. p.i) singularity is one of the form \(S={\text{Spec}} ((R[X]/(h))_{(X,M)})\), where \((R,M)\) is a local, three dimensional regular local ring, essentially of finite type over a field \(k\) of characteristic \(p > 0\), \(X\) an indeterminate, \(h=X^p - g ^{p-1}+f\), with \(f\) and \(g\) in \(M\), \(g \not=0\) (resp. \(f \in M\) and \(g=0\)). In another paper they show that when \(k\) is a DF-field, this type of local uniformization is always possible, concluding the proof of their Main Theorem (see \url{http://hal.archives-ouvertes.fr/hal-00139445}).
In the article being reviewed, the authors use refinements of the original valuation-theoretic approach of Zariski and ramification methods pioneered by Abhyankar, as well as a variety of other algebraic and geometric techniques. As usually in this kind of work, parts of the article are highly technical. However, it is very well written and relatively self-contained. resolution of singularities; local uniformization; ramification; principalization; Galois approximation V.COSSARTand O.PILTANT,\textit{Resolution of singularities of threefolds in positive characteristic I}: \textit{Reduction to local uniformization on Artin-Schreier and purely inseparable coverings}, J. Algebra 320 (2008), no. 3, 1051--1082. http://dx.doi.org/10.1016/j.jalgebra.2008.03.032.MR2427629 Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, \(3\)-folds, Ramification problems in algebraic geometry, Rational and birational maps Resolution of singularities of threefolds in positive characteristic. I: Reduction to local uniformization on Artin-Schreier and purely inseparable coverings | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(K\) be a field, let \(R\) be a two-dimensional regular local subring of \(K\) having \(K\) as field of quotients, let \(\mathfrak m\) be the maximal ideal of \(R\), and let \(\Omega(R)\) be the set of all two-dimensional regular local subrings of \(K\) containing \(R\); these rings shall be called points. The local rings of closed points of \(\text{Bl}_{\mathfrak m}(R)\), the blow-up of \(\text{Spec}(R)\) at its closed point, lie in \(\Omega(R)\), and are called the quadratic transforms of \(R\) or the points in the first neighborhood of \(R\). For any \(S\), \(T \in\Omega(R)\) with \(S\subset T\), the ring \(T\) dominates \(S\), there exists a uniquely determined sequence \(S=:S_0\subsetneqq S_1\subsetneqq \cdots\subsetneqq S_n=:T\) such that \(S_i\) is a quadratic transform of \(S_{i-1}\) which is called the quadratic sequence between \(S\) and \(T\), and the residue field of \(T\) is a finite extension of the residue field of \(S\); its degree shall be denoted by \([T:S]\). For \(S\), \(T\in \Omega(R)\) with \(S\subset T\) one says that \(T\) is proximate to \(S\), denoted by \(S\prec T\), if the discrete valuation ring defined by the order function of the maximal ideal of \(S\) contains \(T\). For each \(f\in R\) we have the (strict) transform \((fR)^S\) of the ideal \(fR\) in \(S\) (for these results which do not use the authors' language, one should confer papers of \textit{J. Lipman} [in: Algebraic geometry and commutative algebra, Vol.\ I, 203--231 (1988; Zbl 0693.13011); in: Commutative algebra: syzygies, multiplicities, and birational algebra, Contemp. Math. 159, 293--306 (1994; Zbl 0814.13016)], or Chapter VII of the book ``Resolution of Curve and Surface Singularities in characteristic zero'' by the reviewer and \textit{J. L. Vicente} [Kluwer Academic Publishers, Dordrecht (2004; Zbl 1069.14001)]; for the language used by the authors one should confer to the first author's book [``Singularities of plane curves'' (Cambridge University Press, Cambridge) (2000; Zbl 0967.14018)]).
A cluster \({\mathcal K}\) with origin \(R\) is a finite subset of \(\Omega(R)\) with \(R\in {\mathcal K}\) and the following additional property: If \(S\in{\mathcal K}\), then all the rings of the quadratic sequence between \(S\) and \(R\) belong to \({\mathcal K}\), also. A pair \(({\mathcal K},\nu)\) where \({\mathcal K}\) is a cluster and \(\nu\colon {\mathcal K}\to \mathbb Z\) is a map, is called a weighted cluster. A weighted cluster is called consistent if one has \(\nu_S-\sum_{T\in{\mathcal K},S\prec T}[T:S]\nu_T\geq0\) for every \(S\in{\mathcal K}\). To every weighted cluster \(({\mathcal K},\nu)\) one can associate a complete ideal \(H_{\mathcal K}\) of \(R\) of finite colength. Also in this more general context one can define for weighted clusters a process called unloading [cf.\ the first author's book, chapter 4, section 4.6] which associates to a weighted cluster \(({\mathcal K},\nu)\) a new weighted cluster \(({\mathcal K'},\nu')\) in such a way that \(H_{\mathcal K}=H_{\mathcal K'}\), and that after a finite number of steps one gets a consistent cluster. The ideal associated to a consistent cluster is the unique complete finite colength ideal \(\mathfrak a\) of \(R\) with \(\text{ord}_S(\mathfrak a^S)\geq \nu_S\) for all \(S\in {\mathcal K}\). An element \(f\in R\) is said to go through \({\mathcal K}\) if \(f\in H_{\mathcal K}\), and \(f\) is said to go sharply through \({\mathcal K}\) if \(\text{ord}_S((fR)^S)=\nu_S\) for all \(S\in {\mathcal K}\) and \(f\) has no singular points outside \({\mathcal K}\) (a point \(S\in\Omega(R)\) is called a non-singular point of \(f\) if \(\text{ord}_S((fR)^S)=1\) and no \(T\in\Omega(R)\) with \(T\supset S\) and \((fR)^T\neq T\) is a satellite of \(S\), i.e. the quadratic sequence from \(S\) to \(T\) contains only one point which is proximate to \(T\)). The authors of the paper under review start with a ring \(R\) of formal power series in two indeterminates over the complex numbers; all what has been said above remains true if one takes the adic completion of all the rings involved. They consider two non-zero elements \(f\), \(g\in R\) which are not units, which satisfy \(\text{ord}_R(f)\leq \text{ord}_R(g)=:n\) and have no common tangent (equivalently, for any quadratic transform \(S\) of \(R\) not both ideals \((fR)^S\), \((gR)^S\) are proper ideals of \(S\)) and they study the family \(h^\lambda:=f+\lambda g\), \(\lambda\in \mathbb C\).
Their main result is theorem \(4.6\) which states: There exists a weighted cluster \({\mathcal T}\) and a finite subset \(M\) of the complex numbers such that, for all \(\lambda\in \mathbb C\setminus M\), , the element \(h^\lambda\) goes sharply through \({\mathcal T}\), and no two such elements have a common point outside \({\mathcal T}\). This result determines -- in the case of convergent power series -- the topological type of \(h^\lambda\) in terms of \(n\) and the singularity of \(f\). For other results in this direction, cf.\ [\textit{T. C. Kuo} and \textit{Y. C. Lu}, Topology 16, 299--310 (1977; Zbl 0378.32001); \textit{H. Maugendre}, C. R. Acad. Sci., Paris, Sér. I 322, No.10, 945--948 (1996; Zbl 0922.32022) and \textit{B. Teissier}, in: Seminaire sur les singularités, 193--221. Publ.\ Math.\ Univ.\ Paris VII 7, Paris (1980; Zbl 0624.32001)]. equisingularity; topological type; pencil; germ of plane curve; weighted cluster Singularities in algebraic geometry, Singularities of curves, local rings, Local complex singularities, Equisingularity (topological and analytic) Perturbing plane curve singularities | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Motivated by results in dimension 2, there is some hope that in general a proper birational map between smooth algebraic varieties over a field of characteristic 0 factorizes into a sequence of smooth blowups, followed by a sequence of smooth blowdowns (strong factorization conjecture). There have been attempts to prove this -- unfortunately containing gaps. The article under review discusses a local version [cf. \textit{D. Cutkosky's} results, ``Local monomialization and factorization of morphisms'', Astérisque. 260 (Paris: Société Mathématique de France) (1999; Zbl 0941.14001)]. The author shows a toric version of local strong factorization.
Theorem 0.2. Let \(\sigma\) and \(\tau\) be nonsingular cones, and let \(v\in\sigma\cap \tau\) be a vector with rationally independent coordinates. Then there exists a nonsingular cone \(\rho\) obtained from both \( \sigma\) and \(\tau\) by sequences of smooth star subdivisions along \(v\).
Using the monomialization theorem of Cutkosky [loc. cit], this is applied to obtain a complete proof of the local factorization conjecture for local rings dominated by a valuation.
Theorem 0.1. Let \(R\subset S\) be regular local rings, essentially of finite type over a field \(k\) of characteristic 0. Assume that \(R\) and \(S\) have a common fraction field \(K\) and \(\nu\) is a valuation on \(K\). Then there exists a local ring \(T\), obtained from both \(R\) and \(S\) by sequences of monoidal transforms along \(\nu\). toric variety; local strong factorization conjecture Karu K.: Local strong factorization of toric birational maps. J. Algebr. Geom 14, 165--175 (2005) Rational and birational maps, Local structure of morphisms in algebraic geometry: étale, flat, etc., Toric varieties, Newton polyhedra, Okounkov bodies Local strong factorization of toric birational maps | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The authors consider an interpolation problem for linear systems \(L=L_{n,d}(m_1,\dots,m_s)\) of hypersurfaces of degree \(d\) in \(\mathbb P^n\), passing through \(s\) points \(P_1,\dots,P_s\) with multiplicities (at least) \(m_i\) at \(P_i\). Such systems have an expected dimension, and the interpolation problem mainly consists of determining the value of \(\dim(L)\), and finding conditions which imply that the value is as expected. Even in the case when the \(P_i\)'s are general points in \(\mathbb P^n\) a complete answer is unknown, for \(n\geq 2\), though there are conjectures and partial results, expecially for the case of points in the plane. When the \(P_i\)'s are located in special position the interpolation problem is widely open. The authors consider the case of points contained in a rational normal curve \(C\). The main result provides the value of the dimension of \(L\) for any \(n,d,m_1,\dots,m_s\) and for a general choice of \(s\) points in \(C\). The formula is expressed recursively in terms of the numerical data. In the case where all multiplicities \(m_i\) are equal to \(2\), the formula determines an Alexander-Hirschowitz type theorem for points on a rational normal curve, which can be applied to the study of some special decompositions of symmetric tensors. The authors also point out that the blow-up \(X_s^n\) of \(\mathbb P^n\) at \(s\) points in the curve \(C\) is a Mori dream space, and the formula for \(\dim(L)\) is connected with the description of the nef cone and the Mori chamber decomposition of the effective cone of \(X_s^n\). polynomial interpolation; linear systems; fat points; rational normal curves; special effect varieties Divisors, linear systems, invertible sheaves, Singularities of surfaces or higher-dimensional varieties, Hypersurfaces and algebraic geometry On linear systems with multiple points on a rational normal curve | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The article gives a short introduction to the problems of local uniformization and local monomialization in their interaction with the general problem of resolution of singularities. Some details are sketched, with references to recent and forthcoming papers of the author. It starts with classical results of Zariski (his fundamental theorem on local uniformization, the theorem on resolution of surface-singularities), illustrates first difficulties, mentions Abhyankar's results on resolution in dimension \(\leq 3\) and Hironaka's famous theorem on the existence of a resolution over a field of characteristic 0. The second part of the paper explains the problem of monomialization (which in general does not have a solution in positive characteristics). Theorem 3.1 [proved in the author's book ``Local monomialization and factorization of morphisms'', Astérisque No. 260 (Paris: Société Mathématique de France) (1999; Zbl 0941.14001)] gives an affirmative answer to a question of Abhyankar on simultaneous resolution along a valuation in characteristic 0. The final section is devoted to several results on monomialization of morphisms. Together with an overview of the proof, the author announces the existence of a global monomialization of a proper dominant morphism from a 3-fold to a surface (in characteristic 0). resolution of singularities; uniformization Global theory and resolution of singularities (algebro-geometric aspects), Rational and birational maps, Singularities in algebraic geometry, Morphisms of commutative rings Local monomialization. | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Given s distinct points \({\mathfrak P}=(p_ 1,...,p_ s)\) and s integers \({\mathfrak D}=(d_ 1,...,d_ s)\) we name ``subscheme of fat points'' relative to \({\mathfrak P}\) and \({\mathfrak D}\), the subscheme Z(\({\mathfrak P},{\mathfrak D})\) defined by the homogeneous ideal \({\mathfrak P}_ 1^{d_ 1}\cap...\cap {\mathfrak P}_ s^{d_ s}\), where \({\mathfrak P}_ i\subset k[x_ 0,x_ 1,x_ 2]\) is the ideal corresponding to \(p_ i\). Let \(S_ t\) be the linear system consisting of all curves of degree \( t\) containing Z(\({\mathfrak P},{\mathfrak D})\) as a subscheme and let \(I\subset {\mathcal O}_{{\mathbb{P}}_ 2}\) be the ideal sheaf of Z(\({\mathfrak P},{\mathfrak D})\). The linear system \(S_ t\) is said to be regular if \(H^ 1({\mathbb{P}}^ 2,I(t))=0\) and it is a classical problem to decide when \(S_ t\) is regular.
The goal of this paper is to fill a gap in a proof of a theorem by \textit{B. Segre} [cf. Atti Convegno internaz. Geometria algebrica, Torino 1961, 15-33 (1962; Zbl 0104.389)]: for ``generic'' points \(p_ 1,...,p_ s\) of \({\mathbb{P}}^ 2\), the linear system \(S_ t\) is regular for \(t\geq \max \{d_ 1+d_ 2-1,[\sum d_ i/2 ]\}\) where \(d_ 1\geq d_ 2...\geq d_ s\). - This is done showing that, fixed D, there is a flat family \(X\to T\) where T is a smooth irreducible quasi-projective variety parametrizing all the schemes of fat points Z(\({\mathfrak P},{\mathfrak D})\), for all \({\mathfrak P}'s\). regularity of linear system; schemes of fat points Paxia G., On flat families of fat points Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves On flat families of fat points | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system We let \(S=\mathbb C[x_{i,j}]\) denote the ring of polynomial functions on the space of \(m\times n\) matrices and consider the action of the group \(\mathrm{GL}=\mathrm{GL}_m\times \mathrm{GL}_n\) via row and column operations on the matrix entries. For a \(\mathrm{GL}\)-invariant ideal \(I\subseteq S\), we show that the linear strands of its minimal free resolution translate via the BGG correspondence to modules over the general linear Lie superalgebra \(\mathfrak {gl}(m|n)\). When \(I=I_{\lambda }\) is the ideal generated by the \(\mathrm{GL}\)-orbit of a highest weight vector of weight \(\lambda \), we give a conjectural description of the classes of these \(\mathfrak {gl}(m|n)\)-modules in the Grothendieck group, and prove that our prediction is correct for the first strand of the minimal free resolution. determinantal thickenings; syzygies; BGG correspondence; general linear Lie superalgebra; Kac modules; Dyck paths Syzygies, resolutions, complexes and commutative rings, Determinantal varieties, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) Syzygies of determinantal thickenings and representations of the general linear Lie superalgebra | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(u : A \rightarrow A'\) be a regular morphism of Noetherian rings and \(B\) an \(A\)-algebra of finite type. Then any \(A\)-morphism \(v : B \rightarrow A'\) factors through a smooth \(A\)-algebra \(C\), that is \(v\) is a composite \(A\)-morphism \(B \rightarrow C \rightarrow A'\). This theorem called General Néron Desingularization was first proved by the second author [Nagoya Math. J. 100, 97--126 (1985; Zbl 0561.14008)]. Later different proofs were given by \textit{M. André} [Cinq exposés sur la désingularisation. Ecole Polytechnique Fédérale de Lausanne (Handwritten manuscript) (1991)], \textit{R. G. Swan} [in: Lectures in algebra and geometry. Proceedings of the international conference on algebra and geometry, National Taiwan University, Taipei, Taiwan, December 26--30, 1995. Cambridge, MA: International Press. 135--192 (1998; Zbl 0954.13003)] and \textit{M. Spivakovsky} [J. Am. Math. Soc. 12, No. 2, 381--444 (1999; Zbl 0919.13009)]. All the proofs are not constructive. In [J. Symb. Comput. 80, Part 3, 570--580 (2017; Zbl 1406.13006)], the authors gave a constructive proof together with an algorithm to compute the Néron Desingularization for 1-dimensional local rings. In this paper we go one step further. We give an algorithmic proof of the General Néron Desingularization theorem for 2-dimensional local rings and morphisms with small singular locus. The main idea of the proof is to reduce the problem to the one-dimensional case. Based on this proof we give an algorithm to compute the desingularization. smooth morphisms; regular morphisms; Néron desingularization Pfister, G.; Popescu, D., \textit{Construction of Neron Desingularization for Two Dimensional Rings}. arXiv:AC/1612.01827 Étale and flat extensions; Henselization; Artin approximation, Local structure of morphisms in algebraic geometry: étale, flat, etc., Regular local rings, Global theory and resolution of singularities (algebro-geometric aspects) Construction of Néron desingularization for two dimensional rings | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \((A,{\mathfrak m})\) be a \(d\)-dimensional local ring. Let \(\{x_ 1, \dots, x_ d\}\) denote a system of parameters of the ring \(A\). The monomial conjecture (MC) [see \textit{M. Hochster}, Nagoya Math. J. 51, 25- 43 (1973; Zbl 0268.13004)], states that \(x_ 1^ t \dots x_ d^ t \notin (x_ 1^{t+1}, \dots, x_ d^{t+1})A\) for all \(t \geq 1\) for all systems of parameters in any local ring \(A\). It is proved by M. Hochster [see the cited paper], that (MC) is true, e.g., whenever \(A\) is an equicharacteristic local ring. Moreover he showed also that (MC) is equivalent to the direct summand conjecture.
In the present paper the authors prove an interesting equivalent condition for (MC) which says: For a local complete intersection ring \(R\) and an ideal \({\mathfrak a}\) consisting of zero divisors the annihilator \(0:_ R {\mathfrak a}\) of \({\mathfrak a}\) is not contained in any parameter ideal of \(R\). This follows by elaborating on local cohomology modules. local ring; monomial conjecture; system of parameters; direct summand conjecture; complete intersection; local cohomology modules Strooker, J. R.; Stückrad, J., Monomial conjecture and complete intersections, Manuscripta Math., 79, 153-159, (1993) Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Complete intersections, Local cohomology and commutative rings, Linkage, complete intersections and determinantal ideals, Local cohomology and algebraic geometry, Curves in algebraic geometry Monomial conjecture and complete intersections | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system This paper presents a generalization of the theorem of \textit{G. Lancaster} and \textit{J. Towber} [J. Algebra 59, 16--38 (1979; Zbl 0441.14013)] describing an ideal of orbits of sums of highest vectors for irreducible representations of a semisimple algebraic group defined over an algebraically closed field of characteristic 0. The Zariski closures of these orbits are called \(S\)-varieties. In their theorem it was assumed that the highest weights are linearly independent. The present paper describes, under certain restrictions, defining relations of the ideals defining \(S\)-varieties. The idea comes from the Plücker relations. \(HV\)-manifolds; \(S\)-manifolds Geometric invariant theory, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups Systems of generators for ideals of \(S\)-manifolds | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A celebrated conjecture due to Segre-Gimigliano-Harbourne-Hirschowitz states that a linear system \(\mathcal{L}(d;m_1, \dots , m_r)\) with base points in general position is spacial if and only if it is \(-1\)-special.
Such a conjecture is known to hold in some cases.
The case \(r\leq 9\) has been solved by \textit{M. Nagata} [Mem. Coll. Sci., Univ. Kyoto, Ser. A 33, 271--293 (1960; Zbl 0100.16801)]. The case \(m_1= \dots = m_r\leq 11\) has been solved, in various stages, by \textit{A. Hirschowitz} [Manuscr. Math. 50, 337--388 (1985; Zbl 0571.14002)], \textit{T. Mignon} [J. Pure Appl. Algebra 151, No. 2, 173--195 (2000; Zbl 0977.14015)], \textit{S. Yang} [J. Algebr. Geom. 16, No. 1, 19--38 (2007; Zbl 1115.14003)], and \textit{M. Dumnicki} and \textit{W. Jarnicki} [J. Symb. Comput. 42, No. 6, 621--635 (2007; Zbl 1126.14008)].
The homogeneous case (\(m_1=\dots = m_r\)) with multiplicity up to 42 has been solved in [\textit{M. Dumnicki}, Ann. Pol. Math. 90, No. 2, 131--143 (2007; Zbl 1107.14007)]. The quasi homogeneous case (\(m_1= \cdots= m_{r-1}\), \(m_r\) arbitrary) was solved for \(m_1=3\) by \textit{C. Ciliberto} and \textit{R. Miranda} [J. Reine Angew. Math. 501, 191--220 (1998; Zbl 0943.14002)], for \(m_1=4\) in [\textit{J. Seibert}, Commun. Algebra 29, No. 3, 1111--1130 (2001; Zbl 0998.14015)], for \(m_1=5\) in [\textit{A. Laface} and \textit{L. Ugaglia}, Can. J. Math. 55, No. 3, 561--575 (2003; Zbl 1057.14015)], and for \(m_1=6\) by \textit{M. Kunte} [Rend. Semin. Mat., Torino 63, No. 1, 43--62 (2005; Zbl 1177.14031)].
The author of the paper under review proves that the Segre-Gimigliano-Harbourne-Hirschowitz conjecture holds for systems of plane curves of a given degree through points in general position with multiplicities at least \(m, \ldots , m, m_0\), where \(m=7,8,9,10\) and \(m_0\) is arbitrary.
The method used in this paper differs form the degeneration method used in almost all the papers mentioned above: the author uses a reduction algorithm together with a direct computation of dimensions of the system. linear systems; Segre-Gimigliano-Harbourne-Hirschowitz conjecture; systems of plane curves Plane and space curves, Computational aspects of algebraic curves Quasi-homogeneous linear systems on \(\mathbb P^2\) with base points of multiplicity 7, 8, 9, 10 | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The author establishes local Weierstraß\ division and preparation theorems in a general framework, which allows him to establish structural properties of Berkovich analytic spaces over a variety of base rings in a unified manner, generalising his results from [\textit{J. Poineau}, La droite de Berkovich sur \(\mathbb Z\). Paris: Société Mathématique de France (SMF) (2010; Zbl 1220.14001)]. These base rings include valued fields, discrete valuation rings and rings of integers in number fields. More precisely speaking, Poineau shows that the local rings of such spaces are noetherian, excellent and, in the case of affine spaces, regular; he also shows that the structural sheaves of such spaces are coherent. Poineau's results generalise and provide independent proofs of the corresponding statements for complex analytic spaces and Berkovich analytic spaces over non-archimedean fields. Poineau, J, Espaces de berkovich sur \(\mathbf{Z}\): étude locale, Invent. Math., 194, 535-590, (2013) Rigid analytic geometry, Global ground fields in algebraic geometry, Analytic algebras and generalizations, preparation theorems, Non-Archimedean analysis, Commutative Noetherian rings and modules, Regular local rings Berkovich spaces on \(\mathbb Z\): local study | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system In earlier papers [see Rev. Roum. Math. Pures Appl. 38, No. 7-8, 569-578 (1993; 807.14003), J. Math. Kyoto Univ. 33, No. 4, 1029-1046 (1993; Zbl 0816.13019), and Math. Ann. 309, No. 4, 573-591 (1997; Zbl 0894.14005)], \textit{R. Achilles} and \textit{M. Manaresi} gave an algebraic proof of the invariance of the Stückrad-Vogel cycle under deformation to the normal cone and proved a local version of this invariance first at a rational and subsequently at a general component of the cycle.
The aim of the present paper is twofold. First there is an alternative approach to the work of Achilles and Manaresi (loc. cit.) by giving an algebraic proof of the invariance of the Stückrad-Vogel cycle under deformation to the normal cone by using the extended Rees ring as a natural deformation space and by refining and extending somewhat their basic theorem which discusses the connection between distinguished and rational components. -- The second aim is to revisit a classical result of \textit{D. G. Northcott} and \textit{D. Rees} [see Proc. Camb. Philos. Soc. 50, 145-158 (1953; Zbl 0057.02601)], which states that the correct number of sufficiently general elements in an ideal of a local ring generates a minimal reduction. -- Finally there is a sketch of a geometrical proof of this classical result by using the Cayley form of a variety. Stückrad-Vogel cycle; deformation to the normal cone; intersection theory; superficial element; Cayley form Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Formal methods and deformations in algebraic geometry, Deformations and infinitesimal methods in commutative ring theory, Algebraic cycles Deformation to the normal cone, rationality and the Stückrad-Vogel cycle | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Over a dozen years ago, I asked Jerzy Weyman to explain the geometric method of finding finite free resolutions to me. His answer was very clear. He convinced me that this is a technique worth learning. Furthermore, I believe that it is magical that both mathematical meanings for the word ``resolve'' come into play in this technique. Suppose \(R\) is the coordinate ring of the algebraic variety \(Y\). For the sake of concreteness, let us assume that \(Y\) is a subvariety of affine \(N\)-space over the algebraically closed field \(\mathbb K\). In this setting, \(R\) is equal to \(P/I\), where \(P\) is a polynomial ring in \(N\) variables over \(\mathbb K\) and \(I\) is the ideal of polynomials which vanish on \(Y\). To employ the geometric method for finding the resolution of \(R\) by free \(P\)-modules, one first resolves the singularity. Let \(Z\to Y\) be a desingularization of \(Y\). Now, \(Z\) is smooth; so, \(Z\) is defined by a regular sequence and the coordinate ring of \(Z\) is resolved by the Koszul complex for this regular sequence. The Koszul complex resolution is the most completely understood and straightforward of all possible resolutions. To resolve the coordinate ring of \(Y\), one ``need only'' push the Koszul complex down to \(Y\) along the desingularization. Of course, the devil is in the details.
This geometric method was introduced by George Kempf in a series of papers which were published in the early 1970's. The first stunning application of this technique was made by Alain Lascoux in 1978. Suppose the variety \(Y\) is the ``determinantal variety'' of all linear transformations from \(\mathbb K^n\to\mathbb K^m\) of rank less than \(r\). (In the present setting, \(Y\) is a subvariety of affine \(N=mn\) space.) Before Lascoux, the resolution of the coordinate ring, \(R\), of \(Y\) was known only when \(r\) is equal to the minimum of \(n\) and \(m\) (this is the maximal order minor case and the resolution is the Eagon-Northcott resolution of 1962) or when \(n=m\) were equal and \(r=n-1\) (in this, submaximal, case, \(R\) is Gorenstein of codimension \(4\) and the resolution is the Gulliksen-Negård resolution of 1972). Lascoux combined the geometric method, the representation theory of the general linear group, and Raoul Bott's theorem about the cohomology of line bundles on homogeneous spaces to resolve the coordinate ring of all determinantal varieties in characteristic zero. The representation theory of general linear group is less complicated in characteristic zero than it is in positive characteristic. Also, Bott's theorem is definitely a characteristic zero theorem. On the other hand, the Eagon-Northcott complex, the Gulliksen-Negård complex, and the Akin-Buchsbaum-Weyman complex of 1981, which resolves the ring defined by submaximal minors for arbitrary \(m\) and \(n\) all are independent of characteristic. It was not known until Mitsuyasu Hashimoto's work in the 1990's that for general configurations of \(m\), \(n\), and \(r\), the Betti numbers of \(R\) really depend on the characteristic of \(\mathbb K\).
The geometric method also applies to Schubert varieties in various homogeneous spaces, determinantal varieties for symmetric and alternating matrices, conjugacy classes in classical Lie algebras, varieties of complexes, and many more situations. Depending upon one's point of view (a geometer might start with a variety and look for the equations; whereas an algebraist might start with the equations) the geometric method allows one to find the coordinate ring of particular varieties, find the equations which vanish on the varieties, prove Cohen-Macaulayness, prove that certain varieties have rational singularities.
It is all here in one place, with one uniform notation throughout: the combinatorics and representation theory of Schur and Weyl modules, an explanation and proof of Bott's theorem, the geometric method laid out very slowly, and oodles of examples. Each chapter contains exercises which extend the material given in the text. Many of these exercises outline a way to re-do, using the geometric method, results which appear in the literature using ad-hoc methods. If you have considered learning this method, but have been intimidated away from pulling the ideas out of the existing published literature, then read this book. The value of having one uniform notation can not be overestimated. If you pull any two papers about Schur modules out of the literature, they will have different conventions about how a Ferrers diagram should be drawn and when a tableau is standard (and it does not matter if the two papers have the same author!). In the long run, the ideas work whether the boxes are stacked on the floor or hang from the ceiling; but in the short run, if one adopts Ian Macdonald's remedy ``Readers who prefer this convention should read this book upside down in a mirror'', then one is likely to suffer eye strain, if not more serious disorientation.
I have one slight caution to readers of Weyman's book. The phrase ``If \(R\) has characteristic zero'' almost always means ``If \(R\) contains a field of characteristic zero''. In particular, the ring of integers is not one of the rings under consideration. In practice, almost all calculations are made over a field and then transferred to an arbitrary ring by way of a base change; so there really is no confusion, unless one picked up one page out of context without knowing the general approach.
The book under review has instantly became the standard reference. I have seen referees' reports that have instructed the author to ``express the results in the language of Weyman's new book''. I am aware of numerous seminars that are reading Weyman's book and finding new applications of its methods. Schur modules; determinantal varieties; Bott's theorem; Schubert varieties; finite free resolutions; desingularization; Koszul complex resolution; resolution of the coordinate ring; Eagon-Northcott resolution; Gulliksen-Negård resolution J. Weyman, \textit{Cohomology of vector bundles and syzygies}, Cambridge University Press, Cambridge U.K. (2003). Syzygies, resolutions, complexes and commutative rings, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Representation theory for linear algebraic groups, Combinatorial aspects of representation theory, Linkage, complete intersections and determinantal ideals, Determinantal varieties, Global theory and resolution of singularities (algebro-geometric aspects), Research exposition (monographs, survey articles) pertaining to commutative algebra, Research exposition (monographs, survey articles) pertaining to algebraic geometry Cohomology of vector bundles and syzygies | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A linear system of degree \(d\) hypersurfaces in \({\mathbb P}^n\) passing through some points with fixed multiplicity is called special if it does not have the expected dimension. The classification of all the special linear systems through double points is due to Alexander and Hirschowitz, and this is the only case completely solved. In the case of linear systems in \({\mathbb P}^2\) many efforts have been made in order to prove the Segre, Harbourne, Gimigliano and Hirschowitz conjecture. In dimension greater than two, the situation is much more complicated and less clear.
In the paper under review, the authors investigate the behaviour of certain special linear systems in \({\mathbb P}^n\) for \(2\leq n\leq 4\), in order to get interesting examples. In particular they study the multiples of the divisors corresponding to the special linear systems listed in the Alexander-Hirschowitz theorem. This analysis leads the authors to propose interesting questions and open problems. linear systems; fat points Laface, A; Ugaglia, L, On multiples of divisors associated to Veronese embeddings with defective secant variety, Bull. Belg. Math. Soc. Simon Stevin, 16, 933-942, (2009) Divisors, linear systems, invertible sheaves, Projective techniques in algebraic geometry On multiples of divisors associated to Veronese embeddings with defective secant variety | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system It is shown that the schematic image of the scheme of Azumaya algebra structures on a vector bundle of rank 4 over any base scheme is separated, of finite type, smooth of relative dimension 13 and geometrically irreducible over that base and that this construction base-changes well. This fully generalizes \textit{C. S. Seshadri}'s theorem [Proc. int. symp. on algebraic geometry Kyoto, 155--184 (1977; Zbl 0142.14005)] that the variety of specializations of \((2\times 2)\)-matrix algebras is smooth in characteristic \(\neq 2\). As an application, a construction of Seshadri [loc. cit.] is shown in a characteristic-free way to desingularize the moduli space of rank 2 even degree semi-stable vector bundles on a complete curve. As another application, a construction of \textit{M. V. Nori} over \(\mathbb{Z}\) [Appendix in loc. cit.] is extended to the case of a normal domain which is a universally Japanese (Nagata) ring and is shown to desingularize the Artin moduli space [\textit{M. Artin}, J. Algebra 11, 532--563 (1969; Zbl 0222.16007)] of invariants of several matrices in rank 2. This desingularization is shown to have a good specialization property it the Artin moduli space has geometrically reduced fibers -- for example this happens over \(\mathbb{Z}\). Essential use is made of Kneser's concept of `semi regular quadratic modules'. For any free quadratic module of odd rank, a formula linking the half-discriminant and the values of the quadratic form on its radical is derived. Clifford algebra; moduli space; semiregular quadratic form; simple module; vector bundle Balaji, T.E. Venkata: Limits of rank 4 Azumaya algebras and applications to desingularization. Proc. Indian Acad. Sci. (Math. Sci.) 112, 485--537 (2002) Vector bundles on curves and their moduli, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) Limits of rank 4 Azumaya algebras and applications to desingularization. | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A system of s distinct lines \(L_ 1,...L_ s\), passing through the origin in the affine space \({\mathbb{A}}_ k^{n+1}\) (k denoting an algebraically closed field), defines a simple curve singularity. The problem of computing the CM type \(t(L_ 1,...L_ s)\) of the local ring at the origin was considered by \textit{A. V. Geramita} and \textit{F. Orecchia} [J. Algebra 70, 116-140 (1981; Zbl 0464.14007)], who mostly dwelled on the case when the ''direction numbers'' \(P_ 1,...P_ s\) of those lines, considered as points of \({\mathbb{P}}^ n\), are in generic s- position. By definition, this means that the \(s\times (_ n^{d+n})\) matrix G(d), obtained by evaluating all degree d monomials on \(n+1\) indeterminates at the homogeneous coordinates of \(P_ 1,...,P_ s\), has maximal possible rank for every \(d\geq 1.-\) The present authors obtain a general formula in the form: \(t(L_ 1,...L_ s)=\sum^{s- 1}_{j=r}\gamma (j)+rk G(s-1)-(_{\quad n}^{n+s-2})\) where the \(\gamma\) (j)'s are - in principle - computable from certain matrices obtained from G by row-transformations (one has also to assume the leading entries of the direction numbers equal to 1, but this is no restriction on the generality); the integer r is the minimal degree of s hypersurface passing through all s points; \(s\geq 2\) and \(n\geq 2.\)
This is used to deduce a useful expression in the case of generic s- position: \(t(L_ 1,...,L_ s)=\gamma (r)+s-(_{\quad n}^{r+n-1}).\) Examples are given to show that for s and n fixed, the type can vary even if the lines are in generic s-position. lines in affine space; Cohen-Macaulay ring; Gorenstein ring; singularity; lines in generic position; CM type; local ring at the origin Baruch, M.; Brown, W. C.: A matrix computation for the Cohen-Macaulay type of s-lines in affine (n + 1) -space. J. algebra 85, 1-13 (1983) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) A matrix computation for the Cohen-Macaulay type of s-lines in affine \((n+1)\)-space | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system In this paper the author explains some new techniques for studying singularities of linear systems, with applications to birational maps between 3-fold Mori fibre spaces, and especially the property of birational rigidity. These techniques are closely related and all have something to do with Shokurov's connectedness principle -- see section 3.2. Though they do not as yet form a coherent method, they are intended to replace the combinatorial study of the resolution graph, started by \textit{V. A. Iskovskikh} and \textit{Yu. I. Manin} [Math. USSR, Sb. 15 (1971), 141-166 (1972); translation from Mat. Sb., Nov. Ser. 86(128), 140-166 (1971; Zbl 0222.14009)]. The author's goal is to provide concise but complete proofs of the known criteria for birational rigidity of 3-fold Mori fibre spaces. Chapter 2 contains a brief exposition of the Sarkisov program. Chapter 3 is a study of singularities of linear systems, based on Shokurov connectedness and inversion of adjunction, and represents the technical core of the paper. In chapter 4 are given several new proofs and generalisations of the rigidity theorem for conic bundles, first known by \textit{V. G. Sarkisov} [Math. USSR, Izv. 17, 177-202 (1981); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 44, 918-945 (1980; Zbl 0453.14017), Math. USSR, Izv. 20, 355-390 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 2, 371-408 (1982; Zbl 0593.14034)]. In chapter 5 is re-proved the Pukhlikov's rigidity criterion [see \textit{A. V. Pukhlikov}, Izv. Math. 62, No. 1, 115-155 (1998); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 62, No. 1, 123-164 (1998; Zbl 0948.14008)] for Del Pezzo fibrations of degree 1 and 2. Chapter 6 represents a review of the main known rigidity theorems for Fano 3-folds: hypersurfaces and complete intersections. As a pattern example, in section 6.2 is proved the [known by \textit{V. A. Iskovskikh}, J. Sov. Math. 13, 815-868 (1980); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 12, 159-236 (1979; Zbl 0415.14025) and by \textit{V. A. Iskovskikh} and \textit{A. V. Pukhlikov}, J. Math. Sci., New York 82, No. 4, 3528-3613 (1996; Zbl 0917.14007)] rigidity of a general smooth 3-fold complete intersection of type (2,3), with the purpose to illustrate the power of the new methods. Throughout the paper are stated various conjectures and open problems. birational rigidity; Sarkisov program; Shokurov connectedness; 3-fold Mori fibre spaces A. Corti, Singularities of linear systems and 3-fold birational geometry, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser. 281, Cambridge University Press, Cambridge (2000), 259-312. \(3\)-folds, Divisors, linear systems, invertible sheaves, Rational and birational maps Singularities of linear systems and 3-fold birational geometry | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system This paper studies the embedded resolution of an algebroid surface over an algebraically closed field of characteristic zero, that is the spectrum of a ring \( K[[X, Y,Z]]/(F)\).
The main combinatorial object associated to \(F\) is Hironaka's characteristic polygon \(\Delta(F)\). The original motivation of this work is: can the combinatorics bound, in some effective sense, the resolution process?
The paper studies in detail the resolution process for prepared equations, that \(F\) is a generic Weierstrass-Tchirnhausen equation, of the form \(Z^n +\sum_{k=0}^{n-2}a_k(X,Y)Z^k\) with \(a_k\) regular in \(X\) of order \(\nu_k=\nu(a_k)\geq n-k\).
The resolution strategy used is the following: (1) if \((Z,X)\) or \((Z, Y )\) are permissible curves, a monoidal transformation centered at them is performed, (2) otherwise, a quadratic transformation. For prepared equations bounds are given for the number of blow-ups needed before the multiplicity drops.
The paper contains many examples. resolution of surface singularities; Newton-Hironaka polygon; equimultiple locus; blowing-up Singularities of curves, local rings, Complex surface and hypersurface singularities Combinatorics and their evolution in resolution of embedded algebroid surfaces | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A germ of a complex analytic variety is quasi-ordinary if there exists a finite projection to the complex affine space with discriminant locus contained in a normal crossing divisor. Some properties of complex analytic curve singularities generalize to quasi-ordinary singularities in higher dimensions, for example the existence of fractional power series parametrization, as well as the existence of some distinguished, {characteristic} monomials in the parametrization.
The paper gives two different affirmative solutions to a problem of Lipman: do the characteristic monomials of a reduced hypersurface quasi-ordinary singularity determine a procedure of embedded resolution of the singularity?
The first procedure builds a sequence of toric morphisms depending only on the characteristic monomials. Along the way characteristic monomials are defined for toric quasi-ordinary hypersurface singularities, and their properties are studied. The second procedure generalizes a method of Goldin and Tessier for plane branches. A key step is the re-embedding of the germ in a larger affine space using certain approximate roots of a Weierstrass polynomial. In the last two sections the two procedures are compared and a detailed example is worked out. singularities; embedded reolution; topological type; discriminant González Pérez, P.D.: Toric embedded resolutions of quasi-ordinary hypersurface singularities. Ann. Inst. Fourier Grenoble 53(6), 1819-1881 (2003) Complex surface and hypersurface singularities, Toric varieties, Newton polyhedra, Okounkov bodies, Modifications; resolution of singularities (complex-analytic aspects) Toric embedded resolutions of quasi-ordinary hypersurface singularities | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The authors show first a vanishing theorem for families of linear series with base ideal being a fat points ideal. They then apply this result in order to give a partial proof of a conjecture raised by the reviewer, Harbourne and Huneke concerning containment relations between ordinary and symbolic powers of planar point ideals.
One of the central problems in the theory of linear series is the study of linear systems of hypersurfaces in projective spaces with assigned base schemes. This problem is related to various other topics, for example to the polynomial interpolation, the Waring problem, the classification of defective higher secant varieties and to the problem of containment relations between ordinary and symbolic powers of ideals.
Given a finite number \(s\) of points \(P_1,\dots, P_s \in \mathbb{P}^n\) and positive integers \(m_1, \dots ,m_s\), let \({\mathcal{L}}={\mathcal{L}}_{n}(t; m_1,\dots, m_s)\) be the linear system of hypersurfaces of \(\mathbb{P}^n\) of degree \(t\) vanishing in the given set of points with prescribed multiplicities. One is interested in determining the dimension of \({\mathcal{L}}\). The virtual dimension of this space
\[
\mathrm{vdim}({\mathcal{L}}):=\binom{n+t}{n}-1-\sum{i=1}^s \binom{n+m_i-1}{n}
\]
arises by assuming that the conditions imposed by the underlying set of points are independent. The expected dimension is defined as
\[
\mathrm{edim}({\mathcal{L}}):=\text{max}\{\mathrm{vdim}({\mathcal{L}}),-1\}.
\]
If the conditions imposed by the assigned points are not linearly independent, the actual dimension of \({\mathcal{L}}\) is greater than the expected one: in that case we say that \({\mathcal{L}}\) is special.
A subscheme \(Z\) of \(\mathbb{P}^n\), defined by an ideal of the form
\[
I_Z={\mathbf m}_{P_1}^{m_1}\cap \cdots \cap {\mathbf m}_{P_s}^{m_s}
\]
where \({\mathbf m}_P\) denotes the maximal ideal of a point \(P \in \mathbb{P}^n\) is called a fat points scheme and the ideal \(I_Z\) is called a fat points ideal. It follows from the long cohomology sequence attached to the twisted structure sequence of \(Z\)
\[
0 \to \mathcal{I}_Z(t) \to \mathcal{O}_{\mathbb{P}^n}(t)\to \mathcal{O}_Z(t) \to 0
\]
that the system \(\mathcal{L}\) is non-special exactly when the cohomology group \(H^1(\mathbb{P}^n,\mathcal{I}_(t))\) vanishes.
The first main result of the paper the following vanishing theorem.
{Theorem A}. Let \(P_1,\dots, P_s\) be \(s\geq 4\) general planar points. Let \(m_1\geq m_2\geq \cdots \geq m_s\geq 1\) be fixed integers. If \(t \geq m_1+m_2\) and \(\mathrm{vdim}({\mathcal{L}}_{2}(t; m_1,\dots, m_s))\geq \frac{1}{2}(3m^2_4-7m_4+2)\) then
\[
h^1(\mathbb{P}^2,\mathcal{O}_{\mathbb{P}^2}\otimes {\mathbf m}_{P_1}^{m_1}\otimes \cdots \otimes {\mathbf m}_{P_s}^{m_s})=0
\]
that is the system \({\mathcal{L}}_{2}(t; m_1,\dots, m_s)\) is non-special.
Moving to the algebraic side, let \(\mathcal{I}\subset \mathbb{C}[\mathbb{P}^n]=\mathbb{C}[x_0, \dots, x_n]\) be a homogeneous ideal. The \(m\)th symbolic power \(\mathcal{I}^{(m)}\) of \(\mathcal{I}\) is defined as
\[
\mathcal{I}^{(m)}=\mathbb{C}[\mathbb{P}^n]\cap \left(\bigcap_{p\in \mathrm{Ass}(\mathcal{I})} \mathcal{I}^m \mathbb{C}[\mathbb{P}^n]_p\right)
\]
where the intersection is taken in the field of fractions of \(\mathbb{C}[\mathbb{P}^n]\). If \(\mathcal{I}\) is a fat points ideal then the symbolic power is simply
\[
\mathcal{I}^{(m)}=\bigcap_{i=1}^s {\mathbf m}_{P_i}^{m\cdot m_i}.
\]
There has been considerable interest in containment relations between usual and symbolic powers of homogeneous ideals over the last two decades. The most general results in this direction have been obtained with multiplier ideal techniques in characteristic zero by Ein, Lazarsfeld and Smith [\textit{L. Ein} et al., Invent. Math. 144, No. 2, 241--252 (2001; Zbl 1076.13501)] and using tight closures in positive characteristic by \textit{M. Hochster} and \textit{C. Huneke} [Invent. Math. 147, No. 2, 349--369 (2002; Zbl 1061.13005)]. Applying these results to a homogeneous ideal \(\mathcal{I}\) in the coordinate ring \(\mathbb{C}[\mathbb{P}^n]\) of the projective space we obtain the following containment statement: \(\mathcal{I}^{(nr)}\subset \mathcal{I}^r\) for all \(r\geq 0\).
There are examples showing that one cannot improve the power of the ideal \(\mathcal{I}\) on the right-hand side of the previous inequality. Nevertheless, it is natural to wonder to what extent this result can be improved; for example, under additional geometrical assumptions on the zero-locus of \(\mathcal{I}\). In particular, if \(\mathcal{I}\) is a fat points ideal, it is natural to wonder for which non-negative integers \(m\), \(r\) and \(j\) there is the containment
\[
\mathcal{I}^{(m)}\subset \mathcal{M}^j\mathcal{I}^r,
\]
where \(\mathcal{M}\) denotes the irrelevant ideal. Harbourne and Huneke suggested the following {
Conjecture} Let \(\mathcal{I}\) ne a fat points ideal in \(\mathbb{P}^n\). Then
\[
\mathcal{I}^{(nr)}\subset \mathcal{M}^{r(n-1)}\mathcal{I}^r,
\]
for all \(r\geq 1\).
This conjecture has been proved recently by \textit{B. Harbourne} and \textit{C. Huneke} [J. Ramanujan Math. Soc. 28A, 247--266 (2013; Zbl 1296.13018)] for general points in \(\mathbb{P}^2\) and by the first author for general points in \(\mathbb{P}^3\).
In this paper, the authors extend these results to a large family of fat points ideals in \(\mathbb{P}^2\). More specifically they show the following theorem.
{Theorem B}. Let \(\mathcal{I}={\mathbf m}_{P_1}^{m_1}\cap \cdots \cap {\mathbf m}_{P_s}^{m_s}\) be a fat points ideal supported on \(s\geq 9\) points in \(\mathbb{P}^2\). If one of the following conditions holds: {\parindent=6mm \begin{itemize} \item[(a)] at least \(s-1\) among the \(m_i\) are equal (almost homogeneous case); \item [(b)] \(m_1\geq \cdots \geq m_s\geq m_1/2\) (uniformly fat case);
\end{itemize}} then the conjecture above holds, that is, there is the containment
\[
\mathcal{I}^{(2r)}\subset \mathcal{M}^r\mathcal{I}^r,
\]
for all \(r\geq 1\). fat points; linear systems; polynomial interpolations; vanishing theorems Dumnicki, M.; Szemberg, T.; Tutaj-Gasińska, H., A vanishing theorem and symbolic powers of planar point ideals, LMS J. Comput. Math., 16, 373-387, (2013) Divisors, linear systems, invertible sheaves, Plane and space curves, Rational and ruled surfaces, Ideals and multiplicative ideal theory in commutative rings, Polynomial rings and ideals; rings of integer-valued polynomials A vanishing theorem and symbolic powers of planar point ideals | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system For classifying singularities, suitable sets of invariants are required. In the case of curve singularities, the invariants one often uses have a valuative nature, since one has a canonically associated finite set of valuations. Thus, the semigroup of values of a plane curve singularity characterizes Zariski's equisingularity type, and the semigroup of the Arf closure (resp. the saturation) characterizes the multiplicity sequence of the resolution process (resp. the equisingularity type of a generic plane projection) for space curve singularities. Saturation has been introduced by Zariski and by Pham-Teissier in a different way. One of the authors gave another definition of saturation which is very appropriate to handle the value semigroups arithmetically [\textit{A. Campillo}, in Singularities, Summer Inst. Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 1, 211-220 (1983; Zbl 0553.14013 and in Singularities, Banach Cent. Publ. 20, 121-137 (1988; see the preceding review)]. As a consequence the semigroups of one dimensional C. M. local rings have a nice structure which is very much simpler than that of the semigroups of plane curves and, even, they have a more natural geometrical interpretation. This nice structure follows from Arf property which is satisfied by the saturated rings. In this communication, we show how the definitions of Arf and saturation can be extended to the relative situation of an arbitrary (local) ring and finitely many discrete valuations of it, in such a way that the actual semigroups also have a nice structure providing reasonable invariants. As an application, one obtains a classification method for singularities having a canonical resolution. classifying singularities; space curve singularities; saturation; value semigroups; semigroups of one dimensional C. M. local rings; canonical resolution Campillo, A.; Delgado, F.; Núñez, C. A.: The arithmetic of arf and saturated semigroups, Appl. rev. Real acad. Cienc. exact. Fís. natur. Madrid 82, No. 1, 161-163 (1988) Singularities of curves, local rings, Singularities in algebraic geometry, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Global theory and resolution of singularities (algebro-geometric aspects) Arithmetic of the Arf semigroups and saturations. Applications | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system [For part I see J. East China Norm. Univ., Nat. Sci. Ed. 1983, No.2, 11- 18 (1983; Zbl 0525.14022).]
Let the base field K be an algebraically closed field of characteristic \(p>2.\) The classification of the reduced irreducible triplets \((G,\rho,V)\) satisfying \(\dim(G)\geq \dim(V)\) was given in part I. In this part II, the author determines which triplets are prehomogeneous vector spaces and discusses their regularity. It turns out that if p is greater then 5, every regular prehomogeneous vector spaces of characteristic 0 induces a regular irreducible prehomogeneous vector space of characteristic \(p\) by reduction mod \(p\). There is one exception if \(p=5\) and there are 6 exceptions if \(p=3\). Moreover, four types of regular reduced irreducible prehomogeneous vector spaces of characteristic \(p>2\) which do not exist in case of characteristic 0 are obtained, they are as follows: \((GL(n),(1+p^ s)\Lambda_ 1,V(n^ 2))\) \((s>0\), \(n\geq 0)\); \((GL(n),\Lambda_ 1+p^ s\Lambda_{n-1},V(n^ 2))\) \((s>0\), \(n\geq 3)\); \((GL(4),\Lambda_ 1+\Lambda_ 2,V(16))\) \((p=5)\); \((GL(1)\times SL(3),\square \otimes (\Lambda_ 1+\Lambda_ 2),V(1)\otimes V(7))\) \((p=3)\). prehomogeneous vector spaces; characteristic p Zhi Jie Chen, A classification of irreducible prehomogeneous vector spaces over an algebraically closed field of characteristic \?. II, Chinese Ann. Math. Ser. A 9 (1988), no. 1, 10 -- 22 (Chinese). Homogeneous spaces and generalizations, Finite ground fields in algebraic geometry A classification of irreducible prehomogeneous vector spaces over an algebraically closed field of characteristic p. II | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A local ring (S,\({\mathfrak n})\) is called a spot over a local ring (R,\({\mathfrak m})\) when S is the localization of a finitely generated ring extension of R. The authors' criterion for spots involves birational domination of d-dimensional local domains (R,\({\mathfrak m})\) by a local domain (S,\({\mathfrak n})\), i.e. \(R\subseteq S\) have the same quotient field and \({\mathfrak m}=R\cap {\mathfrak n}\), and local domains (R,\({\mathfrak m})\) with property L [for ''Lipman'', see \textit{J. Lipmann}, Ann. Math., II. Ser. 107, 151-207 (1978; Zbl 0349.14004)] which means that every d-dimensional normal spot birationally dominating R is analytically irreducible.
Thus the main theorem states that if (R,\({\mathfrak m})\) is a d-dimensional analytically unramified local domain with property L, (S,\({\mathfrak n})\) is either normal or quasi-unmixed and S birationally dominates R, then S is a spot over R. Applications are given for 2-dimensional regular and for excellent local domains, both of which have property L. A similar result is proved for local domains (R,\({\mathfrak m})\) with property \(L_ k\) under which the extension rings now have dimension k. local affine domain; spot over a local ring; analytically unramified local domain 19.Heinzer, W., Huneke, C., Sally, J.D.: A criterion for spots. J. Math. Kyoto Univ. 26(4), 667-671 (1986) Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Local cohomology and algebraic geometry A criterion for spots | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let L be a very ample divisor on a smooth projective surface S, and let K denote the canonical divisor on S. Sommese and Van de Ven have proved that the linear system \(| L+K|\) is base-point free unless \((S,L)=({\mathbb{P}}^ 2,{\mathcal O}(i))\), \(i=1\) or 2, or \((S,L)\) is a scroll. The morphism \(\phi_{L+K}\) defined by \(| L+K|\) is called the adjunction mapping (associated to L). Let \(S\to^{\alpha}\hat S\to^{\beta}{\mathbb{P}}^ n\) be the Stein factorization of \(\phi_{L+K}\). The author concentrates on the structure of the morphism \(\beta\) and proves the following results:
Theorem 1: Suppose that \(\dim(\phi_{L+K}(S))=2\) and let K and S as above. Then \(\beta\) is an embedding except in the following cases: (i) S is obtained by blowing-up \({\mathbb{P}}^ 2\) at 7 points in general position, and either \(S=\hat S\) and \(L=-2K\), or the map \(\alpha: S\to \hat S\) is the blowing-up of \(\hat S\) at one point P and \(L=\alpha^*(-2\hat K)-E\), where \(E=\alpha^{-1}(P)\); (ii) S is obtained by blowing-up \({\mathbb{P}}^ 2\) at 8 points in general position, and \(L=-3K\); (iii) \(S={\mathbb{P}}({\mathcal E})\) is the projectivization of a rank 2 vector bundle \({\mathcal E}\) over an elliptic curve Y, where \({\mathcal E}\) is a non-split extension \(0\to {\mathcal O}_ Y\to {\mathcal E}\to {\mathcal O}_ Y(P)\to 0,\) \(P\in Y\). If \(\zeta ={\mathcal O}_{{\mathbb{P}}({\mathcal E})}(1)\) is the tautological invertible sheaf, then L is numerically equivalent to \(3\zeta\). - One observes that the finite map \(\beta\) is 2-to-1 in case (i), and 3-to-1 in cases (ii) and (iii).
Theorem 2: There exists an effective divisor \(C\in | L|\) which is a smooth hyperelliptic curve if and only if (S,L) belongs to one of the following cases: (a) \(({\mathbb{P}}^ 2,{\mathcal O}(i))\) with \(i=1, 2\) or 3; (b) S is a geometrically ruled surface over a hyperelliptic curve, and the restriction of L to a fibre has degree 2; (c) S is a rational ruled surface, and the restriction of L to a fibre has degree 2; (d) \(S={\mathbb{P}}({\mathcal E})\) is the projectivization of a rank 2 vector bundle \({\mathcal E}\) over an elliptic curve Y, where \({\mathcal E}\) is a non-split extension \(0\to {\mathcal O}_ Y\to {\mathcal E}\to {\mathcal O}_ Y(P)\to 0,\) with \(P\in Y\). - If \(\zeta ={\mathcal O}_{{\mathbb{P}}({\mathcal E})}(1)\) denotes the tautological invertible sheaf, and F is a fibre of \(S\to Y\), then L is numerically equivalent to \(2\zeta +F\); (e) \((S,L)\) is as described in cases (i) and (ii) of theorem 1. In cases (a),(b),(c) and (d), every smooth divisor \(D\in | L|\) is hyperelliptic, but in case (e) the general element of \(| L|\) is not hyperelliptic.
In this paper some bounds are also given for the degree of the fibres of a ruled surface in \({\mathbb{P}}^ n\). It is proved that a hyperelliptic curve C of genus \(g>0\) can be embedded in the rational surface \({\mathbb{P}}({\mathcal O}_{{\mathbb{P}}^ 1}\oplus {\mathcal O}_{{\mathbb{P}}^ 1}(-e))\) if and only if \(e\leq g+1\). If C is a general hyperelliptic curve of genus g and \(e\leq g+1\), then the curves in \({\mathbb{P}}({\mathcal O}_{P^ 1}\oplus {\mathcal O}_{P^ 1}(-e))\) isomorphic to C move in an algebraic family of dimension \(g+6\). hyperelliptic divisors; very ample divisor; canonical divisor; adjunction mapping; degree of the fibres of a ruled surface SERRANO F., ''The adjunction mapping and hyperelliptic divisors on a surface'', J. Reine Angew. Math. 381 (1987), 90--109. Divisors, linear systems, invertible sheaves, Rational and ruled surfaces, Families, moduli, classification: algebraic theory The adjunction mapping and hyperelliptic divisors on a surface | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system This is an ambitious paper which lays the foundations of homological theory of filtration and applies them to study surface singularities. Here a filtration of a local ring (A,m) is a decreasing sequence of non- zero ideals \((F^ n| n\in {\mathbb{Z}})\) such that \(F^ n=A\quad (n\leq 0),\quad F^ iF^ j\subseteq F^{i+j},\quad \cap F^ n=(0),\) and such that the Rees ring \({\mathcal R}=\oplus_{n\geq 0}F^ nT^ n \) is finitely generated over \(A={\mathcal R}_ 0\). For the authors a good filtration is one whose associated graded ring \(G=\oplus F^ n/F^{n+1} \) has good properties (e.g. normal domain with isolated singularity), and from such filtration (which is in general not ideal-adic) they derive properties of A.
In chapter 1, {\S}1 to 3 are devoted to such subjects as the filtered blowing-up \(X=\Pr oj({\mathcal R})\), local cohomology, canonical modules, divisor class group, dualizing complexes. In {\S}4 a criterion for X to be normal with only rational singularities is given. The main result of {\S}5 is: if A is a normal 2-dimensional local ring and G is a domain with isolated singularity, then the (dual graph of the exceptional divisor of) resolution of singularity of Spec(A) is star-shaped.
In chapter 2, conversely, normal 2-dimensional singularities with star- shaped resolution are studied. Such a resolution defines a filtration on A, and its associated graded ring G is compared with another graded ring called Pinkham-Demazure construction. homological theory of filtration; surface singularities; Rees ring; good filtration; filtered blowing-up; local cohomology; canonical modules; divisor class group; isolated singularity; singularities with star-shaped resolution Tomari, M.; Watanabe, K., Filtered rings, filtered blowing-ups and normal two-dimensional singularities with ''star-shaped'' resolution, Publ. Res. Inst. Math. Sci., 25, 681-740, (1989) Deformations and infinitesimal methods in commutative ring theory, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Singularities of surfaces or higher-dimensional varieties, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Étale and flat extensions; Henselization; Artin approximation Filtered rings, filtered blowing-ups and normal two-dimensional singularities with ``star-shaped'' resolution | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(R^p \subseteq R' \subseteq R\) be a tower of commutative rings of characteristic \(p > 0\), where \(R^p= \{ x^p : x\in R \}\). We recall that a set \(\{\phi \}=\{ \phi_1, \ldots, \phi_r \}\) of elements of \(R\) is said to be a \(p\)-basis of \(R\) over \(R'\) if the monomials \(\phi_1^{a_1}\cdots \phi_r^{a_r}\) (\(0\leq a_i \leq p-1\)) are linearly independent over \(R'\) and \(R=R'[\phi_1, \ldots ,\phi_r]\). For instance if \(R\) is regular and local, \(R'\) is local and \(R\) is a finite \(R'\)-module, \(R\) has a \(p\)-basis over \(R'\) if and only if \(R'\) is regular as well as \(R\).
The author works under the following assumptions:
1. \(R=k[x_1, \ldots , x_n]/I\) where \(k[x_1, \ldots ,x_n]\) is a polynomial ring over an algebraically closed field \(k\) of characteristic \(p>0\) (in this setting \(R^p=\) \(k[x_1^p, \ldots , x_n^p]/\) \(I\cap k[x_1^p, \ldots , x_n^p]\));
2. both \(R\) and \(R'\) are regular domains;
3. \(R\) has a \(p\)-basis \(\{ \phi_1, \ldots , \phi_r\}\) over \(R^p\) where \(1 \leq r=\dim R (\leq n)\);
4. \([Q(R'):Q(R^p)]=p^l\) (\(1 \leq l \leq r-1\)) where \(Q(\cdot )\) denotes the field of fractions.
If \(R\) is a polynomial ring, a set of variables is always a \(p\)-basis of \(R\) over \(R^p\), so condition (3) is always satisfied in this case.
Let \(X=\operatorname{Specm}R\) be the non-singular affine variety consisting of the maximal ideals of \(R\) (with the induced Zariski topology). The above hypotheses imply that, given a subset \(\Gamma_{r-l}= \{b_1, \ldots ,b_{r-l}\}\) of elements of \(R\), the set \(M_{\Gamma_{r-l}} \subseteq X\) of the maximal ideals \(\mathfrak{m}\) of \(R\) such that \(\Gamma_{r-l}\) is a \(p\)-basis of \(R_{\mathfrak{m}}\) over \(R'_{\mathfrak{m} \cap R'}\) is an open subset of \(X\). In this paper the author studies the geometry of \(M_{\Gamma_{r-l}}\), assuming that the images of \(db_1, \ldots ,db_{r-l}\) in \(\Omega_{R/R^p}/\mathfrak{m}\Omega_{R/R^p}\) are linearly independent over \(k\) for all maximal ideals \(\mathfrak{m}\) of \(R\).
To this aim the author defines an explicit morphism \(\psi_{\{\phi \}}\), depending on \(\phi\), from \(X\) to the Grassmannian \(\mathbb{G}(l,V_{\{ \phi \}})\) of the subspaces of dimension \(l\) of the \(r\)-dimensional \(k\)-vector space \(V_{\{ \phi \}}=\bigoplus_{i=1}^r k d\phi_i \subseteq \bigoplus_{i=1}^r R d \phi_i = \Omega_{R/R^p}\) (the last equality comes from the fact that \(\{\phi_1, \ldots ,\phi_r \}\) is a \(p\)-basis of \(R\) over \(R^p\)). Similarly, the author defines another morphism \(\tau_{\Gamma_{r-l}}:X \rightarrow \mathbb{G}(r-l,V_{\{ \phi \}})\).
The main result of the paper (Theorem 2.5) states that the set \(M_{\Gamma_{r-l}}\) is equal to
\[
X \setminus p_1(( \cup_{\mathfrak{m}\in X} \{ \mathfrak{m} \} \times \psi_{\{ \phi \}}(\mathfrak{m})) \cap (\cup_{\mathfrak{m}\in X} \{ \mathfrak{m} \} \times \Sigma_1(\tau_{\Gamma_{r-l}}(\mathfrak{m})))),
\]
where \(p_1\) is the first projection of \(X \times \mathbb{G}(l,V_{ \{\phi \}})\), and \(\Sigma_1(\tau_{\Gamma_{r-l}}(\mathfrak{m}))\) denotes the Schubert cycle of \(\mathbb{G}(l,V_{ \{\phi \}})\) consisting in those \(l\)-dimensional vector spaces that meet \(\tau_{\Gamma_{r-l}}(\mathfrak{m})\) in a subspace of dimension at least 1.
Another interesting result is that \(R'\) admits a \(p\)-basis over \(R^p\) provided that \(\psi_{\{ \phi \}}\) is constant (Proposition 2.7). Grassmannian; Kähler differential; \(p\)-basis; Zariski open set Ono T., A note on p-bases of a regular affine domain extension, Proc. Amer. Math. Soc., 2008, 136(9), 3079--3087 Commutative ring extensions and related topics, Varieties and morphisms, Derivations and commutative rings A note on \(p\)-bases of a regular affine domain extension | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(R\) be a commutative Noetherian ring, \(\Phi\) a system of ideals of \(R,\mathfrak a\in\Phi,M\) an arbitrary \(R\)-module and \(t\) a non-negative integer. Let \(\mathcal S\) be a Melkersson subcategory of \(R\)-modules. Among other things, we prove that if \(H^1_\Phi(M)\) is in \(\mathcal S\) for all \(i< t\) then \(H^i_\mathfrak a(M)\) is in \(\mathcal S\) for all \(i< t\) and for all \(\mathfrak a\in\Phi\). If \(\mathcal S\) is the class of \(R\)-modules \(N\) with \(\mathrm{dim}N\le k\) where \(k\ge -1\) is an integer, then \(H^i_\Phi(M)\) is in \(\mathcal S\) for all \(i< t\) if (and only if) \(H^i_\mathfrak a(M)\) is in \(\mathcal S\) for all \(i< t\) and for all \(\mathfrak a\in\Phi\) .As consequences we study and compare vanishing, Artinianness and support of general local cohomology and ordinary local cohomology supported at ideals of its system of ideals at initial points \(i< t\). We show that \(\mathrm{Supp}_R(H^{\mathrm{dim}M-1}_\Phi(M))\) is not necessarily finite whenever \((R,\mathfrak m)\) is local and \(M\) a finitely generated \(R\)-module. \(ETH\)-cofinite modules; local cohomology; Serre subcategories; system of ideals Local cohomology and commutative rings, Commutative Noetherian rings and modules, Local cohomology and algebraic geometry Lower bounds of certain general local cohomology modules | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(\mathbb F\) be an algebraically closed field of characteristic zero and let \({\mathcal M}_n \;(n\geq 2)\) be the set of all \((n\times n)\)-matrices whose entries are elements of \(\mathbb F\). For an element \(A\in {\mathcal M}_n\) and a positive integer \(j\leq n\), \({\text s}_j(A)\) denotes the sum of all principal minors of size \(j\) of the matrix \(A\). The regular map \(\chi: {\mathcal M}_n\to {\mathbb F}^n\) defined by \(A\mapsto ({\text s}_1(A), {\text s}_2(A), \ldots, {\text s}_n(A))\) is called the characteristic map. For a linear subspace \({\mathcal L}\subseteq {\mathcal M}_n\), we define a subset \({\mathcal S}({\mathcal L})\) as follows:
\[
{\mathcal S}({\mathcal L})=\{ A\in {\mathcal M}_n: \chi(A+{\mathcal L}) \;\text{ is not dense in} \;{\mathbb F}^n\},
\]
which is called a singular set related to the chracteristic map \(\chi\).
The author gives a complete characterization of the subspaces \({\mathcal L}\subseteq {\mathcal M}_2\) such that \(\emptyset\neq {\mathcal S}({\mathcal L})\neq {\mathcal M}_2\). Moreover, a complete characterization of the singular sets \({\mathcal S}({\mathcal L})\) in the case of \(n=2\) is given. Finally, a characterization of the \(n\)-dimensional subspaces \({\mathcal L}\subseteq {\mathcal M}_n\) such that \({\mathcal S}({\mathcal L})=\emptyset\) is obtained by their intersections with conjugacy classes. characteristic map; linear subspace; singular sets Vector spaces, linear dependence, rank, lineability, Eigenvalues, singular values, and eigenvectors, Varieties and morphisms, Classical groups (algebro-geometric aspects), Algebraic systems of matrices On linear subspaces of \(\mathcal {M}_n\) and their singular sets related to the characteristic map | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system In 1991 a computational approach to local algebra has been successfully developed and applied to several situations by \textit{T. Mora} in [Discrete Appl. Math. 33, No. 1--3, 161--190 (1991; Zbl 0752.13016)]. One of the seeds of the results described in [loc. cit.] is the paper [\textit{F. Mora}, Lect. Notes Comput. Sci. 144, 158--165 (1982; Zbl 0568.68029)] in which a tangent cone algorithm is presented. The authors of the paper under review reexamine some of Mora's results which are related to constructions of an associated graded ring (Section 2). Then, they present computational experiments about the application of this approach to the localization of the coordinate ring of an affine variety at an arbitrary prime ideal, based on the computer algebra system \texttt{Singular} (section 3). Several of these experiments are exhaustive and, for instance, give rise to some algorithms, such as an algorithm for lifting syzygies and, hence, a resolution. Useful examples are also provided. An implementation of the algorithms is available in the \texttt{Singular} library \texttt{graal.lib} (\texttt{Singular} distribution 4-0-3 onwards). local ring; associated graded algebra; localization; resolution Local rings and semilocal rings, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Computational aspects in algebraic geometry Mora's holy graal: algorithms for computing in localizations at prime ideals | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(P_1, \dots, P_r\in \mathbb{P}^1\times \mathbb{P}^1\) be points in general position. Let \(\pi:X\to \mathbb{P}^1\times \mathbb{P}^1\) be the blowing up at \(P_1, \dots, P_r\), and denote by \(E_1, \dots, E_r\) the exceptional divisors. For \(d, e\geq 0\) and \(m_1, \dots, m_r\) positive itegers let \(\mathcal{L}_{(d, e)}(m_1P_1, \dots, m_rP_r)\) be the complete linear system of the divisor \(dH_1+eH_2-m_1E_1-\dots -m_rE_r\), where \(H_1, H_2\) denote the pullbacks of \(\mathbb{P}^1\times\{a_1\}\) and \(\{a_2\}\times \mathbb{P}^1,\;a_1, a_2\in\mathbb{P}^1\), respectively. The linear system \(\mathcal{L}_{(d,e)}(m_1P_1, \dots, m_rP_r)\) is called special if it does not have the expected dimension. A combinatorial method is proposed to prove the non-speciality of a linear system. As an application, the classification of special linear systems on \(\mathbb{P}^1\times \mathbb{P}^1\) with multiplicities \(\leq 3\) is given. linear systems; curves; surfaces; fat points Lenarcik, T., Linear systems over \(\mathbb{P}^1 \times \mathbb{P}^1\) with base points of multiplicity bounded by three, Ann. Pol. Math., 101, 105-122, (2011) Plane and space curves, Computational aspects of algebraic curves Linear systems over \(\mathbb{P}^{1}\times \mathbb{P}^{1}\) with base points of multiplicity bounded by three | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The paper consists of notes about blowup-based tools for linear system.
Let \(p_1, \dots, p_r\) be general points in \(\mathbb{P}^2\) and let \(m_1, \dots, m_r\) be positive integers. By \(\mathcal{L}(d;m_1, \dots, m_r)\) we denote the system of plane curves of degree \(d\) with multiplicity at least \(m_j\) at \(p_j\), \(j=1, \dots, r\). \(\mathcal{L}(d;m_1^{\times s_1}, \dots, m_r^{\times s_r})\). is a system with repeated multiplicities. The expected dimension of \(\mathcal{L}(d;m_1, \dots, m_r)\) is
\[
\mathrm{edim}(\mathcal{L}(d;m_1, \dots, m_r)=\max \{\mathrm{vdim}(\mathcal{L}(d;m_1, \dots, m_r)), -1 \},
\]
where the virtual dimension, \(\mathrm{vdim}(\mathcal{L}(d;m_1, \dots, m_r))\), is \(\frac{d(d+3)}{2}-\sum_{j=1}^r {m_j+1 \choose 2}\). A system is special if its effective dimension is strictly greater than the expected one.
Recent years have seen significant advances in the understanding of linear systems with imposed multiple points. The case of points in general position is an important case, with several relevant contributions to the open conjectures of Nagata-Biran-Szemberg and Segre-Harbourne-Gimigliano-Hirschowitz. Most of these rely to some extent on semicontinuity and degeneration methods, which often allow setting up induction arguments on the multiplicity or the number of points.
The formalism of blowups has become an essential tool in te study of linear systems with multiple points, especially when using degeneration methods: the geometry of the variety blown up at the imposed points is important; induction arguments often lead to consider points that are not in general position, but ``infinitely near'', i.e. on blowups; useful degenerations are often built by blowing up the total space of some family.
The author aims to overview the set of blowup-based tools that are being used for specializing and degenerating linear systems, with emphasis on clusters of infinitely near points and Ciliberto-Miranda's \textit{blowup and twist}.
The author takes a rather elementary approach, which should serve as a friendly introduction and guide to original research articles. Sometimes full proof are not given or the exposition restricts to particular cases for the sake of simplicity; in such case the author includes references to the existing bibliography. In particular, the author deals only with linear systems of curves on smooth surfaces defined over the field of complex numbers. linear systems; degenerations methods, fat points Divisors, linear systems, invertible sheaves, Rational and birational maps, Fibrations, degenerations in algebraic geometry, Parametrization (Chow and Hilbert schemes) Blowup and specialization methods for the study of linear systems | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(X=Spec(A)\) be an affine (finite type) scheme over an algebraically closed characteristic zero field, and assume that the reductive algebraic group G acts on X. The main result of this paper, the normal linearization theorem, gives criteria to guarantee that X is a G-vector bundle: the necessary hypotheses are that there is a closed G-stable subscheme \(X_ 0\) of X which contains all closed orbits, which is a local complete intersection in X, and into which there is a G-equivariant retraction \(p: X\to X_ 0,\) then X is a G-vector bundle over \(X_ 0.\)
This result is applied in the case of fix pointed actions (i.e. when the projection of X to the categorical quotient X/G induces a bijection from fixed points). If the fixed points are a local complete intersection, then X is a vector bundle over X/G, and if vector bundles are trivial on X/G then X is G-isomorphic to (X/G)\(\times W\), where W is a G-module whose only closed orbit is a unique fixed point, i.e. G is one fix pointed on W. In particular, if G is one fix pointed on X and the fixed point x is regular on X then (X,x) is G-isomorphic to \((T_ x(X),0).\)
These results in turn generalize previous results of Białynicki-Birula and Panyushev which identify certain actions as conjugate to linear actions on vector spaces. The authors also indicate how their results could have applications to the Jacobian conjecture. normal linearization theorem; fix pointed actions; Jacobian conjecture I. Hedén, \textit{Russell's hypersurface from a geometric point of view}, arXiv:1405. 4561v1 (2014). Group actions on varieties or schemes (quotients), Geometric invariant theory, Representation theory for linear algebraic groups Linearizing certain reductive group actions | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(f: X\to \Delta\) be a proper projective morphism with \(X\) smooth and \(\Delta\) an open disk. Let \(D\) be a divisor on \(X\) and set \(U=X\setminus D\). Assume that \(X_0\cup D\) is a divisor with simple normal crossings. This paper centers around the question whether the number \(\nu_{f_U,\lambda}^j\) of Jordan blocks of largest possible size (\(j+1\) if \(0\leq j \leq n\)) for the eigenvalue \(\lambda\) of the monodromy on \(H^j(U_t,\mathbb C)\) can be computed as \(\dim H^j C^\bullet_{f,\lambda}\), where \(C^\bullet_{f,\lambda}\) is a complex determined by the combinatorics of the intersection lattice consisting of the irreducible components of \(X_0\) and their intersections. For \(\lambda=1\) this is true by the theory of limit mixed Hodge structures. An explicit example at the end of the paper shows that this is in general not the case, even when \(f\) is obtained by a desingularisation of a good compactification of an isolated hypersurface singularity. The \(\nu_{f_U,\lambda}^j\) depend on the positions of the singular points in the embedded resolution. Therefore there are no simple combinatorial formulas. The authors construct a complex \(B^\bullet_{f,\lambda}\) with \(B^j_{f,\lambda}\) a direct factor of \(C^j_{f,\lambda}\), such that \(\nu_{f_U,\lambda}^j=\dim H^j B^\bullet_{f,\lambda}\). The problem is the global triviality of certain local systems. For good compactifications of isolated singularities \(\nu_{f_U,\lambda}^n\) can be computed from the Euler characteristic of the complex \(B^\bullet_{f,\lambda}\). The authors give also sufficient conditions for equality of \(B^j_{f,\lambda}\) and \(C^j_{f,\lambda}\). These are in particular satisfied for superisolated surface singularities.
There is a partial generalisation of the formulas for the case of singular total spaces. local monodromy; limit mixed Hodge structure; nearby cycles Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Mixed Hodge theory of singular varieties (complex-analytic aspects), Variation of Hodge structures (algebro-geometric aspects) Number of Jordan blocks of the maximal size for local monodromies | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system An algorithm of resolution of singularities provides a method to eliminate the singularities of an algebraic variety \(X\) by means of a finite sequence of monoidal transformations with precisely defined regular centres. Such algorithms are available in characteristic zero. A natural and difficult problem is to find, given such an algorithm, bounds for the number of transformations necessary to desingularize a variety. The paper reviewed here makes an interesting contribution to the study of this problem. The setting of this paper is that of \textit{basic objects}: quadruples\((W,(J,c)),E\), where \(E\) is a regular variety (over a zero characteristic field), \(J\) a coherent sheaf of \({\mathcal O}_W\)-ideals, \(E\) a finite sequence of divisors with normal crossings and \(c\) a positive integer. A suitable algorithm of resolution for such basic objects rather easily induces one for resolution of algebraic varieties. The algorithm the author considers is that discussed by \textit{H. Hauser} and \textit{S. Encinas} [Comment. Math. Helv. 77, No. 4, 821--845 (2002; Zbl 1059.14022)], which is a variant of one initially proposed by O. Villamayor. Blanco obtains a bound for the number of transformations necessary for resolution in what she calls the non-exceptional monoidal case, namely, for basic objects where \(W={\mathbf A}^n_k=\mathrm{Spec}(R)\), \(R=k[X_1, \ldots ,X_n]\) (\(k\) a zero characteristic field) an \(J\) corresponding to the principal monomial ideal \((X_1^{a_1}\ldots , X_n ^{a_n})\subset R\), where \(a_1 + \cdots +a_n \geq c\). Even this seemingly special case is complicated, and the final answer involves the so-called Catalan numbers. (Not quite surprisingly, since in this situation the algorithm is of a rather combinatorial nature).
The monomial case is relevant, since the algorithm reduces the general case to a monomial situation. Moreover, monomial ideals are useful in the resolution of hypersurfaces which are toric varieties.
The paper contains several examples and concludes with some remarks on the general monomial case. Resolution of singularities; algorithm of resolution; monomial basic object; monoidal transform; complexity; Catalan numbers Global theory and resolution of singularities (algebro-geometric aspects), Singularities of curves, local rings, Computational aspects in algebraic geometry Complexity of Villamayor's algorithm in the non-exceptional monomial case | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(P_1\), \dots, \(P_n\) be a set of \(n\) general points in \(\mathbb P^2\) over an algebraically closed field of characteristic 0 and let \({\mathbf{m}}=(m_1,\dots,m_n)\) be an \(n\)-uple of positive integers. Denoted by \(\mathcal L=\mathcal L_{\mathbf m}\) the linear system of curves of \(\mathbb P^2\) of degree \(d\) passing through each point \(P_i\) with multiplicity \(m_i\), an open problem in algebraic geometry is determining the dimension of the linear system \(\mathcal L\). By Riemann-Roch, the expected dimension \(v\) of \(\mathcal L\) is:
\[
v=\chi(\mathcal L)-1=\frac{d(d+3)}{2}-\sum_{i=1}^n\frac{m_i(m_i+1)}{2}.
\]
We say that \(\mathcal L\) is special if both cohomology groups \(H^0(\mathcal L)\) and \(H^1(\mathcal L)\) are nontrivial. Of course, if \(\mathcal L\) is nonspecial and the linear system is nonempty, the dimension of the linear system \(\mathcal L\) is known and is precisely the expected one.
A conjecture by \textit{B. Harbourne} [in: Algebraic geometry, Proc. Conf., Vancouver/B.C. 1984, CMS Conf. Proc. 6, 95--111 (1986; Zbl 0611.14002)] and \textit{A. Hirschowitz} [Manuscr. Math. 50, 337--388 (1985; Zbl 0571.14002)] states that \(\mathcal L\) is special if and only if \(\text{Bs}(|\mathcal L|)\) contains a \((-1)\)-curve with multiplicity at least two. In the case in which \(m_i=m\) for each \(i\), so that the linear system \(\mathcal L\) is called homogeneous, the conjecture would imply that there are no special linear systems with \(n\geq 9\). This conjecture has been proved by \textit{C. Ciliberto, F. Cioffi, R. Miranda} and \textit{F. Orecchia} [in: Computer mathematics. Proceedings of the sixth Asian symposium (ASCM 2003), Beijing, China, April 17--19, 2003. River Edge, NJ: World Scientific. Lect. Notes Ser. Comput. 10, 87--102 (2003; Zbl 1053.41001)] for all homogeneous linear systems with \(m\leq 20\). With different methods, \textit{M. Dumnicki} and \textit{W. Jarnicki} [J. Symb. Comput. 42, No. 6, 621--635 (2007; Zbl 1126.14008)] and \textit{M. Dumnicki} [Ann. Polon. Math. 90.2, 131--143 (2007; Zbl 1107.14007)] verified the conjecture for \(m\leq 42\).
The method used by Ciliberto, Cioffi, Miranda and Orecchia is based on the idea of \textit{C. Ciliberto} and \textit{R. Miranda} [J. Reine Angew. Math. 501, 191--220 (1998; Zbl 0943.14002) and Trans. Am. Math. Soc. 352, No. 9, 4037--4050 (2000; Zbl 0959.14015)] of specializing the general points of \(\mathbb P^2\) to a line and then to study the degeneration of the linear system \(\mathcal L\).
In this paper the author studies the linear system \(\mathcal L\) specializing the points to an elliptic curve and in the main result of the paper he gives a bound on the dimension of \(\mathcal L\) by the dimension of a simpler linear system in \(\mathbb P^2\). The author provides also some applications of this result, giving, in the case the linear system \(\mathcal L\) is homogeneous and \(1+\tfrac{2mn-6d}{n-9}\) is a positive integer, a method to see if \(\mathcal L\) is nonspecial. multiple points; linear systems: homogeneous linear systems; elliptic curves Petrakiev, Ivan, Multiple points in \(\mathbf {P}^2\) and degenerations to elliptic curves, Proc. Amer. Math. Soc., 0002-9939, 137, 1, 65-71, (2009) Divisors, linear systems, invertible sheaves, Projective techniques in algebraic geometry Multiple points in \(\mathbb P^2\) and degenerations to elliptic curves | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system To any say, normal Noetherian local ring \((R,M,k)\) is attached its semigroup \({\mathcal S}\) of complete (integrally closed) ideals. We study the unique factorization property in semigroups of finitely supported complete ideals as defined by \textit{J. Lipman} [Publ. Math. Inst. Hautes Étud. Sci. 36, 195--279 (1969; Zbl 0181.48903)]. Let \((R,M,k)\) be a regular local ring of dimension three, with \(k\) algebraically closed of characteristic zero and \(X\to\text{Spec}\,R\) be a composition of point blowing ups. In theorems 1 and 2, we give strong geometric necessary conditions for the semigroup \({\mathcal I}^X\) of \(M\)-primary complete ideals such that \(I{\mathcal O}_X\) is locally invertible to have semiunique or unique factorization. We conjecture the converse of the unique factorization version of which we prove a special case. Noetherian local ring; complete ideals Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial), Ideals and multiplicative ideal theory in commutative rings, Picard groups On unqiue factorization in semigroups of complete ideals | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A rank \(\ell\) local system of complex vector spaces on \(\mathbb{P}^1_{\mathbb{C}}\setminus S\) where \(S\) is a collection of \(s\) distinct points, is equivalent to an \(s\)-tuple of matrices in GL\((\ell,\mathbb{C})\) whose product is the identity matrix. Rank \(\ell\) local systems can then be identified with \(\ell\)-dimensional representations. Such a local system is irreducible if the corresponding representation is irreducible. It is called rigid if any other local system with the same conjugacy classes of local monodromies is isomorphic to it.
The author proposes a construction producing irreducible complex rigid local systems on \(\mathbb{P}^1_{\mathbb{C}}\setminus S\) via quantum Schubert calculus and strange duality. These local systems are unitary and arise from a study of vertices in the polytopes controlling the multiplicative eigenvalue problem for the special unitary groups SU\((n)\) (i.e., determination of the possible eigenvalues of a product of unitary matrices given the eigenvalues of the matrices). Strange duals of the simplest vertices of these polytopes are shown to give all possible unitary irreducible rigid local systems. As a consequence one obtains that the ranks of unitary irreducible rigid local systems, including those with finite global monodromy, on \(\mathbb{P}^1_{\mathbb{C}}\setminus S\) are bounded above if one fixes the cardinality \(s\) of the set \(S\) and requires the local monodromies to have orders dividing \(n\) for a fixed \(n\). The author answers a question of N. Katz by showing that there are no irreducible rigid local systems of rank greater than one, with finite global monodromy, all of whose local monodromies have orders dividing \(n\), when \(n\) is a prime number.
The author proves also that all unitary irreducible rigid local systems on \(\mathbb{P}^1_{\mathbb{C}}\setminus S\) with finite local monodromies arise as solutions to the Knizhnik-Zamolodchikov equations on conformal blocks for the special linear group and gives an inductive mechanism for determining all vertices in the multiplicative eigenvalue problem for SU\((n)\). rigid local systems; multiplicative eigenvalue problem; strange duality; quantum Schubert calculus; KZ equations Structure of families (Picard-Lefschetz, monodromy, etc.), Algebraic moduli problems, moduli of vector bundles, Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation), Vector bundles on curves and their moduli Rigid local systems and the multiplicative eigenvalue problem | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system This paper is about a formalization of known procedures to resolve singularities of algebraic varieties defined over fields of characteristic zero. This formalization is presented as a game, called \textit{Stratify}. Probably the name is due to the fact that, after all, a canonical resolution of singularities means to stratify, in a suitable natural way, the singular locus \(S\) of the variety (and those of its transforms) as a union of locally closed regular subvarieties, at each step of the desingularization process we blow up the ``worst'' stratum (which is closed).
At each stage of the game we have a finite weighted graph, which is successively modified by two players \(A\) and \(B\), according to certain rules. It is a rather curious game, because only player \(A\) can win (by reaching a \textit{final stage.} Player \(B\) can only prevent \(A\) from winning, perhaps forever. The game mimics, in a formal or combinatorial way, the different stages in an attempt to resolve singularities of varieties (in characteristic zero, where we may perform induction by using hypersurfaces of maximal contact), specially by taking blowups with permissible centers.
In the paper, after a review of known results on desingularization, the authors explain the game. The list of rules is rather long and complicated. Then the necessary algebro-geometric concepts are briefly reviewed. Next they explain how a resolution process can be translated into the rules of \textit{Stratify}. Actually, the process considered is not one trying to directly resolve singular algebraic varieties, but rather a resolution of pairs \((I,b)\), where \(I\) is a sheaf of nonzero ideals on a regular ambient variety and \(b\) is a natural number (or rather of some closely related objects, the \textit{singularity data}). It is known (and sketched in the paper) that solving this problem implies resolution of varieties. The authors prove that this resolution problem is equivalent to having a winning strategy for ``Stratify'', and finally they show that such a winning strategy is available. singularity; resolution; graph; transversality; singular datum; Rees algebra Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Polynomials in real and complex fields: location of zeros (algebraic theorems), Directed graphs (digraphs), tournaments A game for the resolution of singularities | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system In the desingularization theory (developed by Zariski and Hironaka) one is led to blow up regular centers D contained in the singular locus of a given algebraic variety X. In such a case all points of D have the same \((1)\quad multiplicity,\) \((2)\quad Hilbert\quad polynomial,\) and \((3)\quad Hilbert\) function. The corresponding three conditions are the same if X is a hypersurface, but in general they are different. One can ask for generalizations of the numerical conditions corresponding to (1), (2) and (3). And indeed, it is the main aim of the book under review to develop a general theory concerning these questions in the frame of commutative algebra. In this sense this book may be considered as a special course in commutative algebra. It has nine chapters and ends with a long appendix written by B. Moonen.
The first three chapters develop the basic facts and techniques concerning multiplicities, Hilbert functions, reduction of ideals, generalities about graded rings and blowing ups, characterizations of quasi-unmixed ideals, etc. - In chapter four various notions of equimultiplicity are presented (all of them coinciding in the hypersurface case), while in the next chapter one illustrates how these notions can be used in order to study the behaviour of the Cohen-Macaulay property undergoing a blowing up. - In chapter six one continues the idea of the previous chapter by using these conditions of equimultiplicity in the study of the numerical behaviour of singularities of blowing up singular centers. - In the last three chapters one discusses the local cohomology and the duality of graded rings (somehow in a parallel way with the case of local rings). For example, one studies local rings (A,m) with finite local cohomology \(H^ i_ m(A)\) for \(i<\dim (A)\), with applications to the affine cones over a projective variety; one also studies the Cohen-Macaulay properties of the Rees rings.
The appendix by B. Moonen has three parts: the first one treats the fundamentals of the local complex-analytic geometry, the second one deals with the geometric description of the multiplicity of a complex space- germ as the local mapping degree of a generic projection, and the third one with the theory of compact Stein neighbourhoods and the properties of normal flatness in the analytic case. The book is selfcontained and written with all details needed for a beginner. normal flatness; desingularization; Hilbert polynomial; Hilbert functions; blowing ups; equimultiplicity; local cohomology; Cohen- Macaulay; Rees rings; local complex-analytic geometry Herrmann, M.; Ikeda, S.; Orbanz, U., Equimultiplicity and blowing up. \textrm{An algebraic study, with an appendix by B.\ Moonen}, 3-540-15289-X, xviii+629 pp., (1988), Springer-Verlag, Berlin Multiplicity theory and related topics, Global theory and resolution of singularities (algebro-geometric aspects), Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to commutative algebra, Local cohomology and algebraic geometry, Local analytic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Equimultiplicity and blowing up. An algebraic study. With an appendix by B. Moonen | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The linear system of hypersurfaces \(\mathbb{L}_n(d, m_1,\dots, m_r)\) of degree \(d\) in \(\mathbb{P}^n\) passing trough the points \(p_1,\dots, p_r\) with multiplicities \(m_1,\dots, m_r\), respectively, is called special if its dimension is bigger than the expected dimension. \textit{C. Ciliberto} conjectured [in: 3rd European congress of mathematics, Barcelona, 2000, Vol. I, Prog. Math. 201, 289--316 (2001; Zbl 1078.14534)] that, for general \(p_1,\dots,p_r\), special linear systems correspond to \((-1)\)-curves on the blow-up of \(\mathbb{P}^n\) in \(p_1,\dots, p_r\) associated to a curve in \(\mathbb{P}^n\) through \(p_1,\dots, p_r\) along which the general member of the linear system is singular.
Using the linear system \(\mathbb{L}_3(9,6,4, 4,4,4,4, 4,4,4)\), the authors disprove this conjecture. To show the speciality of this linear system, they apply the Riemann-Roch theorem to the threefold obtained by blowing up \(\mathbb{P}^3\) in nine general points. expected dimension; blowing-up; divisors Antonio Laface and Luca Ugaglia, A counterexample to a conjecture on linear systems on \Bbb P³, Adv. Geom. 4 (2004), no. 3, 365 -- 371. Divisors, linear systems, invertible sheaves, Hypersurfaces and algebraic geometry, Computational aspects of higher-dimensional varieties, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series A counterexample to a conjecture on linear systems on \(\mathbb{P}^3\) | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(R\) be a regular noetherian local ring of dimension \(n\geq 2\) and \((R_i)\equiv R=R_{0}\subset R_{1}\subset R_{2}\subset \dots \subset R_i\subset \dots \) be a sequence of successive quadratic transforms along a regular prime ideal \(\mathfrak p\) of \(R\) (i.e if \(\mathfrak p_i\) is the strict transform of \(\mathfrak p\) in \(R_i\), then \(\mathfrak p_i \neq R_i, i\geq 0\)). We say that \(\mathfrak p\) is maximal for \((R_i)\) if for every non-negative integer \(j\geq 0\) and for every prime ideal \(\mathfrak q_j\) of \(R_j\) such that \((R_i)\) is a quadratic sequence along \(\mathfrak q_j\) with \(\mathfrak p_j \subset \mathfrak q_j\), we have \(\mathfrak p_j = \mathfrak q_j\). We show that \(\mathfrak p\) is maximal for \((R_i)\) if and only if \(V=\cup _{i\geq 0}R_i/\mathfrak p_i\) is a valuation ring of dimension one. In this case, the equimultiple locus at \(\mathfrak p\) is the set of elements of the maximal ideal of \(R\) for which the multiplicity is stable along the sequence \((R_i)\), provided that the series of real numbers given by the multiplicity sequence associated with \(V\) diverges. Furthermore, if we consider an ideal \(J\) of \(R\), we also show that Spec\((R/J)\) is normally flat along Spec\((R/\mathfrak p)\) at the closed point if and only if the Hironaka's character \(\nu ^{*}(J,R)\) is stable along the sequence \((R_i)\). This generalizes well known results for the case where \(\mathfrak p\) has height one [see \textit{B.M. Bennett}, Ann. Math. (2) 91, 25--87 (1970; Zbl 0198.06101)]. 14.Granja, A., Sánchez-Giralda, T.: Valuations, equimultiplicity and normal flatness. J. Pure Appl. Algebra 213(9), 1890-1900 (2009) Valuation rings, Singularities in algebraic geometry, Local structure of morphisms in algebraic geometry: étale, flat, etc. Valuations, equimultiplicity and normal flatness | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(K\) be a field and \(X=(x_{ij})\) a \((m\times n)\)-matrix of indeterminates over \(K\), \(n\geq m\). With \(S=K[x_{ij}]\), \(X\) determines the generic \(S\)-linear map \(\phi:S^n\rightarrow S^m.\) Let \(\text{Spec}R\) be the locus in \(\text{Spec}S\) where \(\phi\) has non-maximal rank: \(R\) is the quotient of \(S\) given by the maximal minors of \(X\), and is the generic determinantal variety.
The classical \(R\)-modules \(M_a=\text{cok}\bigwedge^a_S\phi\) are maximal Cohen-Macaulay and are resolved by the Buchsbaum-Rim complex. In this article the authors prove that the \((M_a)_a\) yields a kind of non-commutative desingularization of the singular variety \(\text{Spec} R\): For \(1\leq a\leq m\) put \(M_a=\text{cok}\bigwedge^a_S\phi\) and \(M=\bigoplus_a M_a\). Then \(E=\text{End}_R(M)\) is maximal Cohen-Macaulay as an \(R\)-module with finite global dimension. That is, \(E\) is a non-commutative desingularization of \(\text{Spec} R\).
If \(m=n\) then \(R\) is the hypersurface \(R=S/(\det\phi)\) and so \(R\) is Gorenstein and the non-commutative desingularization is an example of a non-commutative crepant resolution.
The authors give a description by generators and relations of the non-commutative resolution \(E\) by stating that \(E\) as a \(K\)-algebra is isomorphic to the path algebra \(K\tilde Q\) of some quiver \(\tilde Q\).
The results above are purely algebraic, but are proved by relating them to algebraic geometry. The classical fact that \(\text{Spec} R\) has a Springer type resolution of singularities is frequently used: Define the incidence variety
\[
\mathcal Z=\{([\lambda],\theta)\in\mathbb P^{m-1}(K)\times M_{m\times n}(K)|\lambda\theta=0\}
\]
with projections \(p^\prime:\mathcal Z\rightarrow\mathbb P^{m-1}\) and \(q^\prime:\mathbb Z\rightarrow\text{Spec} R\). The key geometric facts then include: The scheme \(\mathcal Z\) is projective over \(\text{Spec} R\), which is of finite type over \(K\). The \(\mathcal O_{\mathcal Z}\)-module
\[
\mathcal T := p^{\prime\ast}\left(\bigoplus^m_{a=1}\left(\bigwedge^{a-1}\Omega_{\mathbb P^{m-1}}\right)(a)\right)
\]
is a classical tilting bundle on \(\mathcal Z\) , i.e.
(1) \(\mathcal T\) is a locally free sheaf, in particular, a perfect complex on \(\mathcal Z\),
(2) \(\mathcal T\) generates the derived category \(\mathcal D(\text{Qch}(\mathcal Z))\); \(\text{Ext}^\bullet_{\mathcal O_{\mathcal Z}}(\mathcal T, C)=0\) for a complex \(C\) in \(\mathcal D(\text{Qch}(\mathcal Z))\) implies \(C\cong 0\),
(3) \(\text{Hom}_{\mathcal O_{\mathcal Z}}(\mathcal T,\mathcal T[i])=0\) for \(i\neq 0\),
(4) \(M\cong \mathbf{R}q^\prime_\ast\mathcal T\) and
(5) \(E\cong\text{End}_{\mathcal Z}(\mathcal T)\).
These geometric considerations leads to an interesting and important result stating that the variety \(\mathcal Z\) is the fine moduli space for the \(\tilde Q\)-representations \(W\) of dimension vector \((1,m-1,\left(\begin{smallmatrix} m-1\\2\end{smallmatrix}\right),\dots,1)\) that are generated by the last component \(W_m\).
The proofs of the results depends mostly on the explicit computation of the cohomology of certain homogeneous bundles on \(\mathcal P^{m-1}\), determination of higher direct images of twisted bundles of homomorphisms between the modules of differential forms and other technical results. The article is more or less self contained, containing e.g. the construction of the projective tautological Koszul complex. In addition, of interest in itself is a construction of projective resolutions from sparse spectral sequences. This is then used in to construct the non-commutative desingularization \(E\) above, with algebra structure given by the quiverized Clifford algebra and its presentation.
Particularly nice is the treatment of the noncommutative desingularization as a moduli space for representations. It is really interesting to notice that the points in \(\mathcal Z\) corresponding to the simple representations in \(W\) as those lying over the non-singular locus of \(\text{Spec} R\).
The article is strongly recommended to anyone who will understand this level of representation theory in the algebraic geometric view. Be prepared to use a lot of effort to go through all proofs in detail. non-commutative desingularization; sparse spectral sequence; simple representations; tautological Koszul complex; maximal Cohen-Macaulay; quivers; Clifford algebra Abhyankar, S.: Uniformization in a \( p\)-cyclic extension of a two dimensional regular local domain of residue field characteristic \( p\) . Festschrift zur Gedächtnisfeier für Karl Weierstrass 1815 - 1965, Wissenschaftliche Abhandlungen des Landes Nordrhein-Westfalen \textbf{33} (1966), 243-317, Westdeutscher Verlag, Köln und Opladen Noncommutative algebraic geometry, Cohen-Macaulay modules, Global theory and resolution of singularities (algebro-geometric aspects), Riemann-Roch theorems, Rings arising from noncommutative algebraic geometry, Syzygies, resolutions, complexes and commutative rings, Representations of quivers and partially ordered sets Non-commutative desingularization of determinantal varieties. I | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(S=k[x_1, \dots, x_n]\) be the polynomial ring over the field \(k\). Let \({\mathfrak L}\) be a nonzero sublattice of \(\mathbb{Z}^n\) which contains no nonnegative vectors, then its associated lattice ideal in \(S\) is
\[
I_{\mathfrak L} =({\mathbf x}^{\mathbf a} -{\mathbf x}^{\mathbf b} \mid {\mathbf a}, {\mathbf b} \in\mathbb{N}^n\text{ and }{\mathbf a} -{\mathbf b} \in{\mathfrak L})
\]
where monomials are denoted by \({\mathbf x}^{\mathbf a} =x^{a_1}_1 \cdots x_n^{a_n}\) for \({\mathbf a} =(a_1, \dots, a_n)\). The lattice ideal \(I_{\mathfrak L}\) is generic if it is equal to \(({\mathbf x}^{a_1} -{\mathbf x}^{b_1}, {\mathbf x}^{a_2} -{\mathbf x}^{b_2}, \dots, {\mathbf x}^{a_r} -{\mathbf x}^{b_r})\) where none of the \(r\) vectors \({\mathbf a}_i -{\mathbf b}_i\) has a zero coordinate. An algebraic counterpart \(F_{\mathfrak L}\) of the Scarf complex \(\Delta_{\mathfrak L}\) is given. It is shown that \(F_{\mathfrak L}\) is contained in the minimal free resolution of \(S/I_{\mathfrak L}\) and if \(I_{\mathfrak L}\) is generic then \(F_{\mathfrak L}\) is the minimal free resolution of \(S/I_{\mathfrak L}\). This resolution is monomial and does not depend on \(\text{char} (k)\). It is also shown how to recover the Scarf complex of a generic lattice ideal from that of a reverse lexicographic initial ideal. syzygies; toric varieties; polynomial ring; monomials; Scarf complex; generic lattice ideal; lexicographic initial ideal I. Peeva and B. Sturmfels, Generic lattice ideals , J. Amer. Math. Soc. 11 (1998), 363--373. JSTOR: Polynomial rings and ideals; rings of integer-valued polynomials, Toric varieties, Newton polyhedra, Okounkov bodies, Syzygies, resolutions, complexes and commutative rings, Ideals and multiplicative ideal theory in commutative rings Generic lattice ideals | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Author's abstract: In this article, we prove the following generalization of a result of Hartshorne: Let \((S,\mathfrak{n})\) be a regular local ring of dimension 4. Assume that \(x\), \(y\), \(u\), \(v\) is a regular system of parameters for \(S\) and \(a:=xu+yv\). Then for each finitely generated \(S\)-module \(N\) with \(\operatorname{Supp}N=V(aS)\) the socle of \(H^2_{(u,v)S}(N)\) is infinite dimensional. Also, using this result, for any commutative Noetherian complete local ring \((R,\mathfrak{m})\), we characterize the class of all ideals \(I\) of \(R\) with the property that, for every finitely generated \(R\)-module \(M\), the local cohomology modules \(H^i_l (M)\) are \(I\)-cofinite for all \(i\geq 0\). cofinite module; cohomological dimension; local cohomology; Noetherian complete local ring; regular ring Local cohomology and commutative rings, Local cohomology and algebraic geometry, Commutative Noetherian rings and modules Cofiniteness over Noetherian complete local rings | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The paper under review deals with the de Rham cohomology of local systems on \(p\)-adic projective manifolds parametrised by a Drinfeld halfspace. The systems are given by finite dimensional representations of the covering group. Much is already known about them, by previous work of Peter Schneider and others. One of the main tools is the affine cover of the halfspace parametrised by the vertices of the Bruhat-Tits building.
Here the author applies this technique to the monodromy filtration. It is defined by using Drinfeld's semistable model and logarithmic crystalline cohomology. It is identified with another previously known filtration, and as a consequence one obtains a purity result. In this case the filtration is very special as all its nontrivial subquotients have the same dimension.
The second main result of the paper is the degeneration of the Hodge spectral sequence for algebraic local systems, in case Drinfeld's universal \(p\)-divisible group ``comes'' from an abelian use of a Kuga-Satake variety where the problem reduces to cohomology with constant coefficients. Drinfeld space; crystalline cohomology Grosse-Klönne E.: On the p-adic cohomology of some p-adically uniformized varieties. J. Algebr. Geom. 20, 151--198 (2011) \(p\)-adic cohomology, crystalline cohomology On the \(p\)-adic cohomology of some \(p\)-adically uniformized varieties | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system This is the first part of a series of papers devoted to the resolution of singularities (and containing joint work with K. Matsuki, as mentioned in 0.6). The goal is presenting a program towards the construction of a resolution algorithm which works for an algebraic variety over a perfect field in positive characteristic. The author's intention is however, to develop a program working in full generality, i.e. including characteristic 0 as well.
The intended parts of the complete work are:
I. Foundation; the language of the idealistic filtration
II. Basic invariants associated to the idealistic filtration and their properties
III. Transformations and modifications of the idealistic filtration
IV. Algorithm in the framework of the idealistic filtration
Part I (under review) establishes the notion and fundamental properties of the \textit{idealistic filtration}, which is considered the main language of this program.
Chapter 0 starts with a crash course on the existing algorithms in characteristic 0 and introduces the author's program as a ``new approach to overcome the main source of troubles in the language of the \textit{idealistic filtration}, which is a refined extension of such classical notions as the idealistic exponent by Hironaka, the presentation by Bierstone-Milman, the basic object by Villamayor, and the marked ideal by Wlodarczyk.''
Section 0.2.3 introduces the idealistic filtration and mentions some of its distinguished features as there are: leading generator systems as substitutes for hypersurfaces of maximal contact, construction of the strand of invariants through enlargements of an idealistic filtration, saturation and a ``new nonsingularity principle''.
In 0.3 (``Algorithm constructed according to the program'') the author refers to the forthcomimg part IV of the paper as far as termination of the algorithm (in case of positive characteristic) is concerned; he mentions that this question is not yet settled.
In 0.5 a brief account of (mainly references to) the history of the problem is given, as well as hints to recent announcements and approaches.
The remaining about 70 pages of the paper (Part I) contain essentially the ``local'' ingredients of the program. Below follows (a part of) the author's outline, taken from 0.8:
``In Chapter 1, we recall some basic facts on the differential operators, especially those in positive characteristic. Both in the description of the preliminaries and in Chapter 1, our purpose is not exhaustively cover all the material, bu only to minimally summarize what is needed to present our program and to fix our notation.''
``Chapter 2 is devoted to establishing the notion of an idealistic filtration and its fundamental properties. The most important ingredient of Chapter 2 is the analysis of the \(\mathcal D\)-saturation and the \(\mathcal R\)-saturation and that of their interaction. In our algorithm, given an idealistic filtration, we always look for its bi-saturation, called the \(\mathcal B\)-saturation, which is both \(\mathcal D\)-saturated and \(\mathcal R\)-saturated and which is minimal among such containing the original idealistic filtration. The existence of the \(\mathcal B\)-saturation is theoretically clear. However, we do not know a priori whether we can reach the \(\mathcal B\)-saturation by a repetition of \(\mathcal D\)-saturations and \(\mathcal R\)-saturations starting from the given idealistic filtration, even after infinitely many times. The main result here is that the \(\mathcal B\)-saturation is actually realized if we take the \(\mathcal D\)-saturation and then the \(\mathcal R\)-saturation of the given one, each just once in this order. In our algorithm, we do not deal with an arbitrary idealistic filtration, but only with those which are generated by finitely many elements with rational levels. We say they are of r.f.g. type (short for `rationally and finitely generated'). It is then a natural and crucial question if the propery of being of r.f.g. type is stable under \(\mathcal D\)-saturation and \(\mathcal R\)-saturation.''
``In Chapter 3, through the analysis of the leading terms of an idealistic filtration (which is \(\mathcal D\)-saturated), we define the notion of a leading generator system, which \dots plays the role of a collective substitute for the notion of a hypersurface of maximal contact.
Chapter 4 is the culmination of part I, establishing the new nonsingularity principle of the center for an idealistic filtration which is \(\mathcal B\)-saturated. Its proof is given via three somewhat technical but important lemmas, which we will use again later in the series of papers.'' resolution of singulairties; positive characteristic Kawanoue, H.: Toward resolution of singularities over a field of positive characteristic. I. Foundation; the language of the idealistic filtration. Publ. Res. Inst. Math. Sci. 43 (2007), no. 3, 819-909. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Toward resolution of singularities over a field of positive characteristic. I. Foundation; the language of the idealistic filtration | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Algebro-geometric properties are studied for the moduli space of completely reachable pairs arising in the algebraic theory of dynamical systems. Let k be an algebraically closed field of characteristic zero, \(M_{n,m}\) be the set of \(n\times m\) matrices with coefficients from k. Let \((L,N)\in M_{n,n}\times M_{n,m}\). The pair is called completely reachable if its stabilizer is trivial in case of the action of GL(n) transforming (L,N) in \((gLg^{-1},gN)\). Let us consider the set \(V_{n,m}\) of all completely reachable pairs. We define the morphism \(\phi\) by the following diagram: \(V_{n,m}\hookrightarrow M_{n,n}\times M_{n,m}\to M_{n,n}\to^{\alpha}k^ n;\quad V_{n,m}\to^{\pi}M_{n,m}\to^{\phi}k^ n.\) Here \(\pi\) is the geometric factorization according to the action of GL(n), \(\alpha (L)=(C_ 1(L),...,C_ n(L))\), where \(C_ i(L)\) are coefficients of the characteristic polynomials det(T\(\cdot id-L)\). The structure of the fibres \(Z_ a\) of the morphism \(\phi\) is studied.
Main results: The fibre \(Z_ a\) is a normal projective Cohen-Macaulay variety with rational singularities. The set of non-singular points of the fibre \(Z_ a\) is naturally isomorphic to a product of vector bundles over \({\mathbb{P}}^{m-1}\). moduli space of completely reachable pairs; algebraic theory of dynamical systems; characteristic polynomials Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Dynamical systems and ergodic theory, Determinants, permanents, traces, other special matrix functions, Algebraic moduli problems, moduli of vector bundles, Families, moduli, classification: algebraic theory, Linear algebraic groups over arbitrary fields, Classical groups (algebro-geometric aspects) On the algebro-geometric structure of the moduli space of completely reachable systems | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system In a previous paper [\textit{A. Granja} and \textit{M. C. Martínez}, Commun. Algebra 26, No. 7, 2241-2263 (1998; Zbl 0930.14007)] the authors extended Zariski's equisingularity theory to the case of plane curves over a non algebraically closed field of any characteristic.
In the present paper they apply their definition to the following situation: Let \(R\) be an \((n+2)\)-dimensional local noetherian regular ring, \(p\) a height 2 prime ideal of \(R\) such that \(R/p\) is a regular ring and \(f\) a non zero element of \(p\); let \((\underline t)=(t_1, \dots ,t_n)\) be a system of \(p\)-transversal parameters (i.e. the classes of \(t_1, \dots ,t_n\) in \(R/p\) form a system of regular parameters); denote by \(\overline f\) the class of \(f\) in \(R/(\underline {t})R\) and by \(R_{(\underline t)}\) the ring \(R/(\underline t)R\). Then the hypersurface \(V= \text{Spec}(R/fR)\) is said to be equisingular at the maximal ideal \(M(R)\) of \(R\) along \(p\) if there is a system of \(p\)-transversal parameters \((\underline {t'}) = (t'_1, \cdots, t'_n)\) such that the plane curves \(V_{p} = \text{Spec}(R_{p}/fR_{p})\) (the generic section of \(V\) at \(p\)) and \(V_{(\underline t)} = \text{Spec}(R_{(\underline t)}/{\overline f} R_{(\underline t)})\) (the \(p\)-transversal section of \(V\) relative to \((\underline {t'})\)) are equisingular. The aim of the paper is to prove in this situation some natural algebraic properties which hold in the classical case: independence of the choice of \(p\)-transversal parameters, good behaviour by monoidal dilatations, \(\dots\). In particular the authors characterize the singular locus and prove an equisingularity criterion in codimension \(1\), which is similar to the one of \textit{S. S. Abhyankar} [Am. J. Math. 90, 342-345 (1968; Zbl 0159.22601)]. The main results are obtained under the assumption that the prime factorization of \(f\) in \(R\) agrees with that of \(f\) regarded as an element of the completion of \(R\). equisingularity; plane singular curve Global theory and resolution of singularities (algebro-geometric aspects), Singularities of curves, local rings, Plane and space curves Algebraic aspects of Zariski's equisingularity in codimension 1 | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(S:=K[x_0,\dots,x_n]\) be a polynomial ring over an algebraically closed field \(K\), and let \(I\) be a homogeneous ideal of \(S\) defining a subscheme \({\mathfrak X}\) of projective \(n\)-space \(\mathbb{P}^n_K\). The Castelnuovo-Mumford regularity (or simply regularity) of \(I\), \(\text{reg} I\), is defined as follows: If
\[
0\to \bigoplus^{\beta_p}_{j=1} S(-e_{pj}) @>\varphi_p>> \cdots @>\varphi_1>> \bigoplus^{\beta_0}_{j=1}S(-e_{0 j}) @>\varphi_0>> I\to 0\tag{0.1}
\]
is a minimal graded free resolution of \(I\), setting \(e_i:=\max\{e_{ij};\;1\leq j\leq\beta_i\}\), then \(\text{reg} I:= \max \{e_i-i;\;0\leq i\leq p\}\). In other words, \(\text{reg} I\) is the smallest integer \(m\) for which \(I\) is \(m\)-regular, i.e., \(e_{ij}\leq m+i\) for all \(i,j\). When \(I\) is saturated (i.e., when it is the largest ideal defining \({\mathfrak X})\), we call this the regularity of \({\mathfrak X}\). -- The regularity is a numerical invariant of the ideal \(I\) and is, ``an important measure of how hard it will be to compute a free resolution''.
We give an effective method to compute the regularity of a saturated ideal \(I\) defining a projective curve that also determines in which step of a minimal graded free resolution of \(I\) the regularity is attained. Buchberger's syzygy algorithm; polynomial ring; Castelnuovo-Mumford regularity; free resolution Bermejo, I.; Gimenez, P.: On Castelnuovo -- Mumford regularity of projective curves. Proc. amer. Math. soc. 128, 1293-1298 (2000) Syzygies, resolutions, complexes and commutative rings, Polynomial rings and ideals; rings of integer-valued polynomials, Computational aspects of algebraic curves On Castelnuovo-Mumford regularity of projective curves | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \((R, \mathfrak{m})\) be a regular local ring of dimension \(n\) containing a separably closed field \(k\), \(I\subset R\) an ideal and \(A=R/I\) then \(H^a_{\mathfrak{m}} (H^{n-i}_I(R))\) is injective and supported at \(\mathfrak{m}\). Therefore it is a direct sum of \(e(H^a_{\mathfrak{m}}(H^{n-i}_I(R))\) many copies of the injective hull \(E_{R/\mathfrak{m}}\) of \(R/\mathfrak{m}\). \textit{G. Lyubeznik} [Invent. Math. 113, No. 1, 41--55 (1993; Zbl 0795.13004)] proved that this number \(\lambda_{a,i}(A):=e(H^a_{\mathfrak{m}} (H^{n-i}_I(R)))\) does not depend on the choice of \(R\) and \(I\).
It is proved that these invariants can be described as follows. Let \(X\) be a smooth \(k\)--variety and \(Y\subset X\) a closed \(d\)--dimensional subvariety, let \(x\in Y\) and \(A=\mathcal O_{Y,x}\) such that for \(i\neq d\) the modules \(H^{n-1}_{[Y]}(\mathcal O_X)\) are supported in \(x\). Then for \(\lambda_{a,d}(A)=\lambda_{0,d-a+1}(A)\) for \(a\neq d\) and \(\lambda_{d,d}(A)=\lambda_{0,1}(A)+1\) and all other \(\lambda_{a,i}(A)\) vanish, \(2\leq a\leq d\). Moreover,
\[
\lambda_{a,d}(A)-\delta_{a,d}=\left\{ \begin{matrix} \dim_{\mathbb F_p} H_{\{x\}}^{d-a+1}(Y_{\text{ét}}, \mathbb F_p) & \text{ if } \text{char}(k)=p>0\\ \dim H_{\{x\}}^{d-a+1}(Y_{\text{an}},\mathbb C) & \text{ if } k=\mathbb C\qquad\qquad\;. \end{matrix}\right.
\]
local cohomology; characteristic \(p\); perverse sheaves Blickle, Manuel, Lyubeznik's invariants for cohomologically isolated singularities, J. Algebra, 308, 1, 118-123, (2007) Local cohomology and algebraic geometry, Étale and other Grothendieck topologies and (co)homologies Lyubeznik's invariants for cohomologically isolated singularities | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system This paper is divided in two sections. In the first one the following theorem is proved:
Let \((R,{\mathfrak m}, K)\) be an excellent henselian Cohen-Macaulay local ring, with isolated singularity \(\text{Spec}(R)\) containing a field such that \(K\) is perfect or \([K:K^p] < \infty\) if \(\text{ch} (K)=p<0\). Let \(x=\{x_1, \dots, x_s\}\) be a system of parameters in \(R\) and \(R_t=R/(x^t_1, \dots, x_s^t)\), \(t\in\mathbb{N}\). Let \(L_{R_2}\) be the category of all finitely generated \(R_2\)-modules such that \(((x_1, \dots, x_{j-1}) P:x_j)_P=(x_1, \dots, x_j)P\), for all \(j=1, \dots, s\). Let \(\text{MCM} (R)\) be the category of maximal CM-modules of \(R\). Then the base change functor \(F_1: \text{MCM} (R)\to \text{Mod} (R_1)\), induces a bijection from the isomorphism classes of \(\text{MCM} (R)\) to the isomorphism classes of \(R_1\)-modules of the type \(x_1\dots x_sP\) for \(P\in L_{R_2}\).
In the second section it is shown that the isomorphism classes of maximal Cohen-Macaulay \(R\)-modules and the isomorphism classes of modules of \(L_{R_2}\) can be seen as orbits of an affine group acting on a Grassmannian variety when the residue field of \(R\) is algebraically closed. isomorphism classes of finitely generated modules; isomorphism classes of maximal Cohen-Macaulay modules; isolated singularity; Grassmannian variety Popescu, D.: Maximal Cohen--Macaulay modules over isolated singularities. J. algebra 178, 710-732 (1995) Cohen-Macaulay modules, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Grassmannians, Schubert varieties, flag manifolds, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Maximal Cohen-Macaulay modules over isolated singularities | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(k\) be an algebraically closed field of characteristic zero, \(X\) a non-singular variety over \(k\) and \(\mathcal{I}\) a non-zero ideal sheaf over \(X.\) The problem of principalization of \(\mathcal{I}\) is to get a proper birational morphism \(\phi:Y\to X\), by means of a finite sequence of monoidal transforms such that \(Y\) is non-singular and \(\mathcal{IO}_{Y}\) is locally principal. In the paper under review, the author gives an algorithm for principalization of a locally monomial ideal sheaf on a 3-fold. Then he shows how this algorithm can be used to construct a toroidalization of a locally toroidal morphism of 3-folds. principalization; toroidalization; resolution of morphisms \(3\)-folds, Divisors, linear systems, invertible sheaves, Global theory and resolution of singularities (algebro-geometric aspects), Local structure of morphisms in algebraic geometry: étale, flat, etc. A principalization algorithm for locally monomial ideal sheaves on 3-folds with an application to toroidalization | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Zariski's local uniformization, a weak form of resolution of singularities, implies that every valuation ring containing \(\mathbf{Q}\) is a filtered direct limit of smooth \(\mathbf{Q}\)-algebras. Given an immediate extension of valuation rings \(V\subset V'\) containing \(\mathbf{Q}\) we show that \(V'\) is a filtered direct limit of smooth \(V\)-algebras. This corrects a paper of us [23] where we thought that we may reduce to the case when the value groups are finitely generated. For this correction we use an infinite tower of ultrapowers construction that rests on results from model theory. Valuation rings, Extension theory of commutative rings, Ultraproducts and related constructions, Singularities in algebraic geometry Néron desingularization of extensions of valuation rings (with an appendix by Kęstutis Česnavičius) | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system We study local systems arising from flat line bundles over topologically trivial families \(U\to S\) of hyperplane complements in \(\mathbb P^n\). Imposing some genericity condition on the monodromy, one knows that fiberwise the cohomology of the local system is concentrated in the middle dimension and is computed by the Aomoto complex, a subcomplex of global differential forms on a good compactification \(\pi:X\to S\) with logarithmic poles along \(D'=X \setminus U\).
The families \({\mathcal A}'\) considered are obtained by fixing a configuration \({\mathcal A}\) of hyperplanes and moving one additional hyperplane. The line bundle is the structure sheaf, endowed with the connection \(d_{\text{rel}} +\omega\), for a logarithmic relative differential form \(\omega\). In this situation we construct the Gauß-Manin connection \(\Delta\) on \(R^n\pi_* (\Omega^\bullet_{X/S} (\log D'), d_{\text{rel}}+\omega)\). We show that these sheaves are free. Using the combinatorics of \({\mathcal A}'\) we give a basis for these sheaves and an algorithm to express the connection \(\nabla\) in this basis. The corresponding matrix depends too much on the combinatorics of the arrangement to be given in a closed form, but we illustrate the method with some examples.
These results can be seen as a generalization of the hypergeometric functions. arrangements of hyperplanes; hyperplane complements; global differential forms; Gauß-Manin connection; hypergeometric functions Herbert Kanarek, Gauß-Manin connection arising from arrangements of hyperplanes, Illinois J. Math. 44 (2000), no. 4, 741 -- 766. Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Transcendental methods, Hodge theory (algebro-geometric aspects), Projective techniques in algebraic geometry Gauß-Manin connection arising from arrangements of hyperplanes | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system This paper generalizes \textit{M. M. Kapranov}'s [Adv. Sov. Math. 16(2), 29--110 (1993; Zbl 0811.14043)] well-known construction of \(\overline{\mathcal{M}}_{0,n}\) as the Chow quotient \((\mathbb{P}^1)^d /\!/_{\text{Ch}} \text{SL}_2\) by considering the set
\[
U_{d,n} := \left\{ (p_1, \dots, p_n) \in (\mathbb{P}^d)^n \,\mid\, p_i \text{ are distinct points lying on a rational normal curve} \right\}
\]
and its closure \(V_{d,n} := \overline{U_{d,n}}\) in \((\mathbb{P}^d)^n\). This closure is shown to consist of all configurations of (possibly coinciding) points that lie on a degeneration of a rational normal curve. The author shows that for \(d \leq n-3\) the Chow quotient \(V_{d,n} /\!/_{\text{Ch}} \text{SL}_{d+1}\) is isomorphic to \(\overline{\mathcal{M}}_{0,n}\), and that for any effective linearization \(L\) there exists a morphism to the GIT quotient \(\varphi: \overline{\mathcal{M}}_{0,n} \to V_{d,n} /\!/_L \text{SL}_{d+1}\), extending the isomorphism \(\mathcal{M}_{0,n} \widetilde{\to}\; U_{d,n} / \text{SL}_{d+1}\). Every effective linearization can be encoded by a tuple \(L = (x_1, \dots, x_n)\) of rational numbers with the conditions that \(0 \leq x_i \leq 1\) and \(\sum_{i=1}^n x_i = d+1\), and the morphism \(\varphi\) is shown to factor through the corresponding moduli space \(\overline{\mathcal{M}}_{0,L}\) of \(L\)-weighted pointed stable curves that was constructed in [\textit{B. Hassett}, Adv. Math. 173, No. 2, 316--352 (2003; Zbl 1072.14014)].
Moreover it is shown that if \(L = (\frac{d+1}{n}, \dots, \frac{d+1}{n})\) is the unique \(S_n\)-invariant linearization and \(\mathcal{L}_d\) denotes the GIT polarization on \(V_{d,n} /\!/_L \text{SL}_{d+1}\), then \(\varphi^* \mathcal{L}_d\) spans the same ray in \(N^1(\overline{\mathcal{M}}_{0,n})\) as a certain conformal blocks divisor studied previously in [\textit{M. Arap} et al., Int. Math. Res. Not. 2012, No. 7, 1634--1680 (2012; Zbl 1271.14034)]. Results from that paper imply that for \(d = 1, \dots, \lfloor \frac{n}{2} \rfloor - 1\) the line bundles \(\varphi^* \mathcal{L}_d\) span distinct extremal rays of the symmetric nef cone of \(\overline{\mathcal{M}}_{0,n}\). On the other hand, using the classical Gale transform the author shows that \(\varphi^* \mathcal{L}_d\) and \(\varphi^* \mathcal{L}_{n - d - 2}\) span the same ray in \(N^1(\overline{\mathcal{M}}_{0,n})\), so the spaces \(V_{d,n} /\!/_L \text{SL}_{d+1}\) and \(V_{n-d-2,n} /\!/_L \text{SL}_{n-d-1}\), with \(L\) the respective symmetric linearizations, have isomorphic normalizations. conformal blocks; moduli space of stable pointed rational curves; Chow quotient; GIT quotient; weighted pointed stable curves; Gale duality Giansiracusa, N., Conformal blocks and rational normal curves, J. Algebraic Geom., 22, 4, 773-793, (2013) Families, moduli of curves (algebraic), Geometric invariant theory, Parametrization (Chow and Hilbert schemes) Conformal blocks and rational normal curves | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Consider a smooth complex projective variety \(X\) of dimension \(n \geq 2\) and an ample base point free linear system \(| V| \) on \(X\). It can occur that there exists a point \(x \in X\) such that when imposing the elements of \(| V| \) to pass through \(x\) we get only reducible or non--reduced elements. The point \(x \in X\) is defined as a bad point for the pair \((X,V)\).
In a previous work by the same authors [Manuscr. Math. 112, No. 2, 197--219 (2003; Zbl 1045.14002)] is shown that bad points can appear only when \(\dim(X)=2\) and \(| V| \) is not very ample. The goal of the paper under review is to generalize this kind of results. There are two natural ways to generalize them. One possibility is to see points as reduced zero--schemes of length one and then study zero-schemes for which any element of the linear system of elements of \(| V| \) through them is reducible or non-reduced. Another possibility is to consider a point as a zero-dimensional linear space and then study linear spaces on \(X\) verifying the same property.
In the former the following problem appears: study the minimal length, if finite, \(b\) (respectively \(b_0\)) of a zero-scheme (respectively reduced zero-scheme) \(Z\) for which the described property holds: any element of \(| V| \) through \(Z\) is reducible or non-reduced. A related question on the number of independent conditions imposed by \(Z\) on \(| V| \) is considered and a notion of suitable pair \((V,Z)\) appears (see Section 2 for details). As a generalization of the previous quoted result, in Rmk. 23 and Thm. 22 it is shown that \(\dim(X) \leq b_0+1\) and \(\dim(X) \leq b+1\), the second assertion under the assumption of the existence of a suitable pair. Stronger bounds are proven when further (positivity) hypotheses are imposed on \(| V| \). Examples showing the sharpness of all the bounds are presented. In the latter it is shown (see Section 4) that bad linear spaces can only appear in codimension 2 and are inherited by hyperplane section, then they do not occur for very ample systems. linear systems; bad locus; zero-schemes Besana, Gian Mario; Di Rocco, Sandra; Lanteri, Antonio: Higher order bad loci, J. pure appl. Algebra 211, 414-427 (2007) Divisors, linear systems, invertible sheaves, Embeddings in algebraic geometry Higher order bad loci | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system In this nicely written paper, the authors provide a general vanishing result for the first cohomology group of affine smooth complex varieties with values in some rank-one local systems, and they apply it in the context of a certain family of complex reflection line arrangements in the projective plane.
Main Theorem. Let \(S\) be a smooth proper complex variety of dimension \(N \geq 2\), and let \(D = \sum_{i=1}^{n} D_{i}\) be a divisor with \(D_{i}\) irreducible. Let \(I \subset \{1,\dots,n\}\) be a subset and consider the corresponding subdivisor \(D' = \sum_{i \in I} D_{i}'\). Let \(\mathcal{L}\) be a rank-one local system on \(U = S \setminus D\). Denote the monodromy around \(D_{i}\) by \(t_{i}\in \mathbb{C}^{\times}\). Assume that
i) \(D\) is normal crossing along \(D'\), that is, for any \(p \in D'\), \(D\) is normal crossing around \(p\);
ii) \(t_{i}\neq 1\) for \(i \in I\);
iii) \(U'=S\setminus D'\) is an affine variety.
Then \(H^{1}(S\setminus D, \mathcal{L}) = 0\).
As a corollary, the authors prove the following nice result.
Proposition. The monodromy eigenspaces \(H^{1}(F,\mathbb{C})_{\lambda}\) of the Milnor fiber cohomology for the monomial arrangement
\[
\mathcal{A}(m,m;3) : (x^{m}-y^{m})(y^{m}-z^{m})(x^{m}-z^{m}) = 0
\]
are trivial if \(\lambda^{3} \neq 1\), for any \(m \geq 3\). affine varieties; cohomologies; vanishing results; line arrangements; local systems Relations with arrangements of hyperplanes, Vanishing theorems in algebraic geometry, Classification of affine varieties, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Milnor fibration; relations with knot theory A vanishing result for the first twisted cohomology of affine varieties and applications to line arrangements | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The authors' introduction is a rather complete exposition of the results, the body of the article containing explicit definitions and proofs of the results. So this review is just a compressed version of the introduction and verbatim expressions occur.
The main issue of the article is to generalize standard results about non-commutative resolutions of quotient singularities for finite groups to arbitrary reductive groups. This involves the statement of the standard resolution and the proof of their generalization.
Throughout, \(k\) is an algebraically closed field of characteristic \(0\), all rings are defined over \(k\), for an algebraic group \(G\), \(\widehat G\) denotes the set of isomorphism classes of irreducible \(G\)-representations.
For a normal noetherian domain \(S\), a non-commutative (crepant) resolution (NC(C)R) is an \(S\)-algebra of finite global dimension of the form \(\Lambda=\text{End}_S(M)\) where \(M\) is a non-zero finitely generated reflexive \(S\)-module. It is called crepant if \(S\) is Gorenstein and \(\Lambda\) is a maximal Cohen-Macaulay \(S\)-module.
The author remarks that the HPD-dual of a smooth projective variety, as defined by Kuznetsov, often is a NCR of a singular variety.
The NCCRs are the best NCRs as they have good homological properties. There is a connection to ordinary (commutative) crepant resolutions in algebaic geometry, and it is often possible to pass from one to the other by to a NCCR assigning a crepant resolution as a GIT moduli space of representations, and conversely a crepant resolution might give a NCCR obtained as \(R=\text{End}_X(T)\) for a tilting bundle \(T\) on a variety \(X\).
Also, under mild conditions, a NC(C)R yields a categorical crepant resolution of singularities: If \(S\) is a finitely generated integrally closed \(k\)-algebra with \(\Lambda=\text{End}_S(M)\) a NCR, then \(\Lambda\) is smooth as a DG-algebra. Also, if \(M\) is a projective \(\Lambda\)-module, the functor \(D(S)\rightarrow D(\Lambda):N\mapsto M\overset {L}\otimes_S N\) is fully faithful and yields a categorical resolution of singularities of \(S\). Moreover, if \(\Lambda\) is a NCCR then the categorical resolution is crepant.
In the following, a twisted NC(C)R of index \(u\) behaves as a NC(C)R except that it is generically a central simple algebra of index \(u\) rather than a matrix ring.
When a finite group \(G\) acts on a smooth affine variety \(X\) and \(U\) is a finite-dimensional \(G\)-representation, then \(M(U)=(U\otimes k[X])^G\) is called the corresponding \(k[X]^G\)-module of covariants. The abelian category \(\text{mod}(G,k[X])\) of \(G\)-equivariant finitely generated \(k[X]\)-modules has a projective generator \(U\otimes k[X]\) with \(U=\oplus_{V\in\widehat G}V\), and it follows that \(\text{mod}(G,k[X])\) is equivalent to \(\text{mod}(\Lambda)\) where \(\Lambda=\text{End}_{G,k[X]}(U\otimes k[X])=(\text{End}(U)\otimes k[X])^G=M(\text{End}_k(U))\) so that \(\Lambda\) is a \(k[X]^G\) algebra of covariants. Because \(\text{mod}(G,k[X])\) has finite global dimension, \(\text{gl\; dim} \Lambda<\infty\) and so \(\Lambda\) is a non-commutative resolution of \(k[X]^G=k[X\slash G]\) in a weak sense. If no element of \(G\) fixes a divisor then \(k[X]/k[X]^G\) is étale in codimension one so that \(\Lambda=\text{End}_{k[X]^G}(M(U))\), and then \(\Lambda\) is NCR of \(k[X]^G\). For \(V\) to be a NCCR one also needs \(k[X]^G\) to be Gorenstein, and that is obtained when \(X=\text{Spec}SW\) for a representation \(W\) and \(G\subset\text{SL}(W)\). For general reductive groups, especially when \(G\) is not finite, the generalization is more involved. However, this is handled by the authors.
The main body of the article looks for non-commutative resolutions of quotient singularities given by algebras of covariants. The first main theorem states that when \(G\) is a reductive group acting on a smooth affine variety \(X\), there exists a finite-dimensional \(G\)-representation \(U\) containing the trivial representation such that \(\Lambda=M(\text{End}(U)\otimes k[X])^G\) have \(\text{gl\;dim}<\infty\).
One has that \(\Lambda=M(\text{End}(U))\) is a \(k[X]^G\)-algebra, finitely generated as \(k[X]^G\)-module and with \(k[X]^G\subset Z(\Lambda)\) so that \(\Lambda\) is a weak non-commutative resolution of \(k[X]^G=k[X\slash G]\). As \(U\) contains a trivial direct summand, \(M(U)\) is a projective left \(\Lambda\)-module, one has that \(\text{End}_\Lambda(M(U))=k[X]^G\), and so one gets that if \(G\) is a reductive group acting on a smooth affine variety \(X\), \(k[X]^G\) has a categorical resolution of singularities given by an algebra of covariants.
Correspondingly, when \(X\) is projective, the authors prove a GIT-version of the main result, that is, for given \(G,\; X,\;\mathcal L\), there exists a finite-dimensional \(G\)-representation \(U\) containing the trivial representation such that the coherent sheaf of algebras \(\Lambda=M^{ss}(\text{End}(U))\) on \(X^{ss}\slash G\) has finite global dimension when restricted to affine opens.
A finite group \(G\) is said to be acting generically on a smooth affine variety \(X\) if \(X\) contains a point with closed orbit and stabilizer, and if \(X^s\subset X\) is the locus of points with that property, then \(\text{codim}(X-X^\ast)\geq 2\). Then a \(G\)-representation \(W\) is generic if \(G\) acts generically on \(\text{Spec}SW\cong W^\ast\). In this case, if \(G\) acts generically on \(X\), there exists a finite-dimensional \(G\)-representation \(U\) containing the trivial representation such that \(\Lambda=\text{End}_{k[X]^G}(M(U))\) is a NCR for \(k[X]^G\). Applying the corresponding techniques to abelian reductive groups, the authors prove that for \(S\subset\mathbb Z^n\) a finitely generated commutative positive normal semigroup, for \(n\in\mathbb N,\;n\gg 0\), the \(k[S]\)-module \(M=k[\frac{1}{n}S]\) defines a NCR for for \(k[S]\). This confirms that in characteristic \(p\) Frobenius twists provide canonical NCRs.
After the development of this theory, with the deep results indicated above, it is applied to some specific quotient singularities.
The first is the determinantal varieties and the result that the variety \(Y_{n,h}\), \(n<h\), of \(h\times h\)-matrices of rank \(\leq n\), \(k[Y_{n,h}]\) has a NCCR.
Considering Pfaffian varieties, letting \(Y_{2n,h}^-\), \(2n<h\), be the variety of skew-symmetric \(h\times h\)-matrices of rank \(\leq 2n\), then for odd \(h\), \(k[Y_{2n,h}^-]\) has a NCCR.
For determinantal varieties for symmetric varieties, for \(t<h\), let \(Y^+_{t,h}\) be the variety of symmetric \(h\times h\)-matrices of tank \(\leq t\). If \(t\) and \(h\) have opposite parity, then \(k[Y^+_{t,h}]\) has a NCCR. If the parity is equal, then \(k[Y^+_{t,h}]\) has a twisted NCCR of index \(2^{\lfloor h/2\rfloor}\).
Finally, the results for non-commutative resolutions for \(\text{SL}_2\)-invariants are stated, and trace rings are proven to be modules of covariants in a series of cases.
The introduction, which is rather exhaustive, ends by stating a number of sufficient conditions for the existences of NC(C)Rs, possibly twisted, for quotient singularities. Thus the necessary machinery is included and this article is a complete introduction to noncommuative resolutions, with a lot of nice examples. non-commutative resolutions; non-commutative crepant resolutions; quotients of reductive groups; HPD dual; DG algebra; twisted NC(C)R; tilting algebra; tilting module Špenko, Š.; Van den Bergh, M., Non-commutative crepant resolutions for some toric singularities I, (2017), preprint Actions of groups on commutative rings; invariant theory, Geometric invariant theory, Derived categories and associative algebras Non-commutative resolutions of quotient singularities for reductive groups | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The paper is concerned with the socalled ``minimal resolution conjecture'' for \(s\geq n+1\) distinct points \(P_ i\) in \({\mathbb{P}}_ k^ n\) (k algebraically closed), not contained in a hyperplane, saying that there exists a non-empty open subset of \(({\mathbb{P}}_ k^ n)^ s\) consisting of sets of points in generic position with the same numerical resolution of the coordinate ring A of the set \(\{P_ 1,...,P_ s\}\). The terminus technicus ``s points in generic position'' is defined via the Hilbert function of A. Now, a minimal graded free resolution for the homogeneous ring A was given by \textit{A. Lorenzini} [see J. Pure Appl. Algebra 60, No.3, 273-288 (1989; Zbl 0702.13013)]. And the conjecture has been solved for \(n=2, 3\) and also for any n if s is chosen in a proper way. The authors compute in the main theorem 2 certain Betti-numbers under specific assumptions on s getting in this way the expected numerical invariants for the last part of the mentioned resolution. This gives a new approach to several known results (see theorems 7 and 8). Moreover, combining their main theorem with results of Green and Lazarsfeld, they can show the conjecture for \(s=n+4\) points in \({\mathbb{P}}^ n\). minimal resolution conjecture; points in generic position; Hilbert function [CRV1] Cavaliere, M.P., Rossi, M.E., Valla, G.: On the resolution of points in generic position. Commun. Algebra19 (4), 1083--1097 (1991) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Relevant commutative algebra On the resolution of points in generic position | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(X\) be a smooth projective complex manifold. Then it is a fundamental problem to understand the behavior of a multiple linear system, and this problem has been studied by many authors. The purpose of this paper is to give effective versions of some well-known theorems on multiple linear systems for \(\dim X=2\). The main result of this paper is the following:
Let \(A\) be a nef and big divisor on a smooth projective complex surface \(X\), let \(T\) be any fixed divisor, and let \(k\) be a nonnegative integer. Assume that either \(n>k+\mathcal{M}(A,T)\), or \(n\geq \mathcal{M}(A,T)\) when \(k=0\) and \(T\sim K_{X}+\lambda A\) for some rational number \(\lambda\), where \(\mathcal{M}(A,T):=((K_{X}-T)A+2)^{2}/4A^{2}-(K_{X}-T)^{2}/4\). Suppose that there exists a zero dimensional subscheme \(\Delta\) on \(X\) with minimal degree \(\deg\Delta \leq k\) such that it does not give independent conditions on \(| nA+T| \). Then there is an effective divisor \(D\not=0\) containing \(\Delta\) such that \(TD-D^{2}-K_{X}D\leq k\) and \(DA=0\).
By direct applications of the above result for various \(T\), we obtain the effective version of known theorems. projective complex surfaces; \(k\)-very ampleness; pseudo-effective divisor; nef divisor Tan S L. Effective behavior of multiple linear systems. Asian J Math, 2004, 8: 287--304 Divisors, linear systems, invertible sheaves, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Riemann-Roch theorems, Surfaces and higher-dimensional varieties Effective behavior of multiple linear systems | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(Q\subseteq \mathbb Z^d\) be an affine semigroup. An irreducible resolution of a module \(M\) over the semigroup ring \(k[Q]\) is an exact sequence \(0\to M\to W^0\to W^1\to \cdots\) with \(W^i=\bigoplus_{j=1}^{\mu_i} k[Q]/W^{ij}\) where each \(W^{ij}\) is an irreducible monomial ideal of \(k[Q]\). The irreducible resolution is called linear if \(W^i\) is pure of Krull dimension dim\((M)-i\) for all \(i\geq 0\).
In the current paper, the author establishes that, for a radical monomial ideal \(I\) in a normal semigroup ring \(k[Q]\), there is a unique minimal irreducible resolution
\[
0\to k[Q]/I\to W^0\to W^1\to \cdots
\]
by modules \(W^i\) of the form \(\bigoplus _jk[F_{ij}]\), where the \(F_{ij}\) are (not necessarily distinct) faces of \(Q\). That is, \(W^i\) is a direct sum of quotients of \(k[Q]\) by prime ideals. This paper characterizes Cohen-Macaulay quotients \(k[Q]/I\) as those whose minimal irreducible resolutions are linear. The proof exploits a graded ring-theoretic analogue of the Zeeman spectral sequence, thereby also providing a combinatorial topological version involving no commutative algebra. The characterization via linear irreducible resolutions reduces to the Eagon-Reiner theorem by Alexander duality when \(Q=\mathbb N^d\).
The author notes that the methods involved here should have applications beyond this work and discusses some possibilities in the final section. Cohen-Macaulay quotients; normal semigroup ring; irreducible resolution; monomial ideal; Zeeman spectral sequence; Eagon-Reiner theorem Miller, E., \textit{Cohen-Macaulay quotients of normal semigroup rings via irreducible resolutions}, Math. Res. Lett., 9, 117-128, (2002) Syzygies, resolutions, complexes and commutative rings, Cohen-Macaulay modules, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), General theory of spectral sequences in algebraic topology, Toric varieties, Newton polyhedra, Okounkov bodies, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Semigroup rings, multiplicative semigroups of rings Cohen-Macaulay quotients of normal semigroup rings via irreducible resolutions. | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(M\to V\) be a resolution of a two-dimensional singularity (V,p). The author considers a local version of the canonical ring, \(S=\oplus S_ n\quad with\) \(S_ n=\Gamma (M,{\mathcal O}(nK))\), set of n pluri-canonical forms on M. The fact that S is generated by \(S_ i\), \(i=1,...,n\), was conjectured by M. Reid, and proved by Shepherd-Barron.
main reslt (also conjectured by M. Reid): The algebra S is generated by \(S_ 1\oplus S_ 2\oplus S_ 3\). This reslt is sharp in the sense that there are examples for which \(S_ 1\oplus S_ 2\) does not generate S. The author also gives a characterization of the cases where \(t: S_ 2\otimes S_ 2\to S_ 4\) is surjective, and on the other hand two examples for which \(S_ 1\otimes S_ 3\) is necessary. The reslt is: t is surjective if and only if there are no minimally elliptic cycle of \(square\quad -1.\) In general the number of these cycles is not greater than \(\dim (S_ 4)/Im(t)\). The different proofs are based on the study of graphs of resolutions, fundamental cycles and Du Val curves. resolution; canonical ring; fundamental cycles H. B. LAUFER, Generation of 4-pluricanonical forms for surface singularities, Amer. J. Math., 109 (1987), pp. 571-589. Zbl0628.14031 MR892599 Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects) Generationof 4-pluricanonical forms for surface singularities | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(S=k[x_0,\ldots, x_n]\) be a polynomial ring over an infinite field \(k\). The authors present a method for bounding the Castelnuovo-Mumford regularity of an \(S\)-module \(M\) in terms of the regularities of modules that form an approximation system for \(M\). This method is inspired by a paper by \textit{A. Conca} and \textit{J. Herzog} [Collect. Math. 54, No. 2, 137--152 (2003; Zbl 1074.13004)] which concerns the regularity of products of linear ideals. In the paper under review, the regularity of systems of ideals generated by linear ideals in \(S\) is studied. As consequence we can deduce Sturmfels' conjecture: ``If \(I\) is the vanishing ideal of \(d\) subspaces, then the Castelnuovo-Mumford regularity of \(I\) is at the most \(d\)'', and we obtain a result proved by \textit{H. Derksen} and \textit{J. Sidman} [Adv. in Math. 172, (2), 151--157 (2002; Zbl 1040.13009)] which regards the regularity of ideals of an arrangement of linear subspaces of projective space. A bound on the regularity of the module of derivations tangent to a hyperplane arrangement is proved. syzygies; subspace arrangements; invariant theory; approximation system; hyperplane arrangement; Castelnuovo-Mumford regularity Derksen, H., Sidman, J.: Castelnuovo-Mumford regularity by approximation. Adv. Math. 188, 10--123 (2004) Syzygies, resolutions, complexes and commutative rings, Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Actions of groups on commutative rings; invariant theory, Polynomial rings and ideals; rings of integer-valued polynomials, Ideals and multiplicative ideal theory in commutative rings, Relevant commutative algebra Castelnuovo-Mumford regularity by approximation | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system In [\textit{R. Bédard} and \textit{R. Schiffler}, Represent. Theory 7, 481--548 (2003; Zbl 1060.17001)] one gives the characterization of the orbits of representations of quivers of type \(A\) with Zariski closures which are rationally smooth. In the paper under review the author continues these investigations and studies the local rational smoothness of these closures. He obtains a description of the orbits with the property that the projectivization of their Zariski closures are rationally smooth.
The main idea of the method is to use that the change of basis between canonical, and PBW-basis of the positive part of the quantized enveloping algebra of type \(A_n\) has a geometric interpretation in terms of local intersection cohomology of some affine algebraic varieties, namely the Zariski closures of orbits of representations of a quiver of type \(A_n\). Then the author applies some methods already used in his paper with Bédard, and some results from there. Comparing the results of both papers, it turns out that the only non-smooth, projectively rationally smooth orbit closures are of type \(A_2\) and \(A_3\). projective representations of quivers; varieties of representations; rational smoothness; quantum groups; quantized enveloping algebra Quantum groups (quantized enveloping algebras) and related deformations, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Singularities in algebraic geometry, Representations of quivers and partially ordered sets, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Projective rational smoothness of varieties of representations for quivers of type \(A\) | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system For \(P_1,\ldots,P_s\) points on a curve, the definition of the Weierstrass semigroup \(H=H(P_1,\ldots,P_s)\) at these points is closely related to the canonical divisor of the curve. In this paper, the author extends the definition of \(H\) to any linear system \(\Gamma\) [cf. \textit{C. Carvalho} and \textit{F. Torres}, Des. Codes Cryptogr. 35 (2), 211--225 (2005; Zbl 1070.94029)]. Let \(G:={\mathbb N}_{0}^n\setminus H\). In the case \(s=1\), \(\ell\in G\) if and only if there is a divisor \(D\in \mu\) such that \(v_P(D)=\ell-1\). The main result in this paper is the computation of a sharp lower bound on \(\# G\) which depends of \(n\) and the dimension of \(\Gamma\) only. If the characteristic of the base field is 0 or \(p>2g-2\), there is a Zariski open set in \(X^n\) where the lower bound holds true; otherwise, this property is not known. Weierstrass semigroups of several points; linear systems; hyperelliptic curves Carvalho C.: On \({\mathcal V}\)-Weiertsrass sets and gaps. J. Algebra \textbf{312}, 956-962 (2007). Riemann surfaces; Weierstrass points; gap sequences On \(\mathcal V\)-Weierstrass sets and gaps | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(X\) be a scheme of finite type over a field \(k\). Let \(D_1, \dots, D_h\) be effective divisors having simple normal crossings on \(X\). A simple constructive proof (working over fields of any characteristic) is given for the existence of a finite sequence of blow-ups at non-singular centers principalizing the ideal \(I_{D_1}+\dots+I_{D_h}\), i.e. if \(\pi: X'\to X\) is the composite of these blow-ups, then the ideal \(\sum I_{D_i}\mathcal O_{X'}\) is a principal ideal defining a divisor with simple normal crossings. In particular it is proved that the centers can always be chosen to be the intersection of two of the prime divisors in the support of \(\sum D_i\). Principalization of monomial ideals is one step in the constructive resolution of singularities. Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: \textsc{Singular} 3-1-6--A Computer Algebra System for Polynomial Computations. http://www.singular.uni-kl.de (2012) Global theory and resolution of singularities (algebro-geometric aspects), General commutative ring theory, Rational and birational maps A simple algorithm for principalization of monomial ideals | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(V\) be a finite-dimensional vector field over an arbitrary ground field. In the paper under review a description of pairs of subspaces in \(V\) of fixed dimensions \(k\) and \(l\) up to the action of the group \(B\subset\text{GL}(V)\) of nondegenerate upper-triangular matrices is obtained. In other words, the author describes \(B\)-orbits in the direct product \(X=\text{Gr}(k,V)\times \text{Gr}(l,V)\) of two Grassmannians. This description is similar to the Schubert decomposition for Grassmannians or to the Ehresmann-Bruhat decomposition for full flag varieties. The combinatorial description of \(B\)-orbits in \(X\) may be found in [\textit{P.~Magyar, J.~Weyman} and \textit{A.~Zelevinsky}, Adv. Math. 141, No. 1, 97--118 (1999; Zbl 0951.14034)]; the proof there is based on quiver theory. The description given in the present paper uses only elementary linear algebra.
The closures of \(B\)-orbits in \(X\) may be viewed as analogues of Schubert varieties in Grassmannians. The singularities of Schubert varieties admit nice desingularizations constructed by Bott and Samelson. In this paper, the author constructs a similar desingularization for the closures of \(B\)-orbits in \(X\). Grassmannian; spherical variety; desingularization Grassmannians, Schubert varieties, flag manifolds, Vector spaces, linear dependence, rank, lineability Desingularizations of Schubert varieties in double Grassmannians | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Authors' abstract: ``We study the vanishing cycles of a one-parameter smoothing of a complex analytic space and show that the weight filtration on its perverse cohomology sheaf of the highest degree is quite close to the monodromy filtration so that its graded pieces have a modified Lefschetz decomposition. We describe its primitive part using the weight filtration on the perverse cohomology sheaves of the constant sheaves. As a corollary we show in the local complete intersection case that 1 is not an eigenvalue of the monodromy on the reduced Milnor cohomology at any points if and only if the total space and the singular fiber are both rational homology manifolds. Also, we introduce quasi-semistable degenerations and calculate the limit mixed Hodge structure by constructing the weight spectral sequence. As a corollary we show non-triviality of the space of vanishing cycles of the Lefschetz pencil associated with a tensor product of any two very ample line bundles except for the case of even-dimensional projective space where two has to be replaced by three.'' vanishing cycles; mixed Hodge module; smoothing of a singularity Dimca, A., Saito, M.: Vanishing cycle sheaves of one-parameter smoothings and quasi-semistable degenerations. arXiv:0810.4896v2 (2009) Mixed Hodge theory of singular varieties (complex-analytic aspects), Deformations of complex singularities; vanishing cycles, Fibrations, degenerations in algebraic geometry, Variation of Hodge structures (algebro-geometric aspects) Vanishing cycle sheaves of one-parameter smoothings and quasi-semistable degenerations | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(\mathcal O=k[[x,y]]\) be the complete local ring of a plane curve with two branches over an algebraically closed field \(k\) of characteristic \(0\), let \(S\subseteq \mathbb Z^2_+\) be the semigroup of values defined by the normalization map \(\mathcal O\to k[[t_1]]\times k[[t_2]]\). The Apery set of \(S\) with respect to \(\alpha\in S\) is defined as \(\Omega_\alpha=\{\beta\in S\mid \beta-\alpha\notin S\}\). Define \(\Omega_\alpha^0=\{\beta\in \Omega_\alpha\mid \beta\text{ maximal}\}\) and inductively \(\Omega_\alpha^i=\{\beta\in \Omega_\alpha \setminus \bigcup_{j=0}^{i-1}\Omega_\alpha^j\mid \beta \text{ maximal}\}\), here \(\beta <\beta'\) if \(\beta_i <\beta'_i\) for \(i=1,2\). One obtains \(\Omega_\alpha=\bigcup_{i=0}^{m-1}\Omega_\alpha^i\). Let \(v(x)=:e\in S\) be the smallest element and \(\mathcal O'=k[[x, \frac{y}{x}]]\) be the blow up of \(\mathcal O\). Assume that \(\mathcal O\) and \(\mathcal O'\) are local rings. It is proved that \(\Omega_e^i=(\Omega')^i_e + (m-1-i)e\). This result is the basis of an algorithm to compute the multiplicity tree using the semigroup and vice versa. Finally the multiplicity trees of plane curve singularities with two branches are characterized. semigroup; multiplicity tree Barucci, V.; D'Anna, M.; Fröberg, R., The apery algorithm for a plane singularity with two branches, Beiträge zur Algebra und Geometrie, 46, 1-18, (2005) Singularities of curves, local rings, Multiplicity theory and related topics, Valuations and their generalizations for commutative rings, Plane and space curves The Apery algorithm for a plane singularity with two branches | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(R\) be a regular local ring, and let \( A=R/(x)\), where \(x\) is any nonunit of \(R\). We prove that every minimal free resolution of a finitely generated \(A\)-module becomes periodic of period 1 or 2 after at most \(\dim A\) steps, and we examine generalizations and extensions of this for complete intersections. Our theorems follow from the properties of certain universally defined endomorphisms of complexes over such rings. complete intersection; periodicity of minimal free resolution; regular local ring; matrix factorization D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), no. 1, 35-64. Homological methods in commutative ring theory, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Complete intersections, Complexes, Regular local rings Homological algebra of a complete intersection, with an application to group representations | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system For Part I, cf. J. Algebr. Geom. 7, No. 3, 539-567 (1998; Zbl 0956.14016).
Let \({\mathfrak A}\) be the moduli scheme over \(\mathbb Z [1/n]\) of principally polarized abelian varieties with level \(n\) structure, whose generic fiber can be described as a Shimura variety. Given an ordinary moduli point \(x \in {\mathfrak A}\) in characteristic \(p\), a local linearity property at \(x\) can be formulated in terms of the Serre-Tate group structure on the formal deformation space, that is, the formal completion of \({\mathfrak A}\) at \(x\). In the case of characteristic zero, such a property is closely linked to a differential-geometric linearity property. In this paper the author proves that an irreducible algebraic subvariety of \({\mathfrak A}\) is a Shimura subvariety if, locally at an ordinary point \(x\), it is formally linear. As an application, he reformulates Oort's conjecture on subvarieties \(Z \hookrightarrow {\mathfrak A}\) with a dense collection of CM-points and proves it in a special case. Shimura varieties; abelian subvarieties; Oort's conjecture; characteristic \(p\); principally polarized abelian varieties; CM-points B. Moonen, ''Linearity properties of Shimura varieties. II,'' Compositio Math., vol. 114, iss. 1, pp. 3-35, 1998. Modular and Shimura varieties, Subvarieties of abelian varieties, Variation of Hodge structures (algebro-geometric aspects), Arithmetic aspects of modular and Shimura varieties Linearity properties of Shimura varieties. II | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(S = \{p_1, \dots, p_s\}\) be a set of distinct points in \({\mathbb{P}}^n = {\mathbb{P}}^n_K\) and let \(\wp_1,\dots, \wp_s\) be the associated homogeneous prime ideals in the polynomial ring \(R := K[x_0, \dots , x_n]\), where \(K\) is an algebraically closed field. Given positive integers \(m_1,\dots ,m_s\) we denote by \(Z :=\sum^s_{i=1} m_i\wp_i\) the \(0\)-dimensional subscheme of \({\mathbb{P}}^n \) defined by the saturated ideal \(I_Z := \wp_1^{m_1}\cap\dots\cap \wp_s^{m_s}\). We denote by \(Z_{\mathrm{red}} := \sum^s_{i=1} p_i\) the support of \(Z\) and by \(w(Z) := \sum^s_{i=1}m_i\) its weight.
The regularity index \(\mathrm{reg}(Z)\) of \(Z\) is the smallest positive integer \(d\) such that \(h_1({\mathbb{P}}^n, I_Z(d)) = 0\) or, equivalently, \(h_1({\mathbb{P}}^n,{L}_{n,d}(m_1,\dots ,m_s)) = 0\) where \({L}_{n,d}(m_1,\dots ,m_s)\) is the linear systems of the degree-\(d\) hypersurfaces of \({\mathbb{P}}^n\) passing through each point \(p_i\) with multiplicity at least \(m_i\), for all \(d\geq 0\). This number corresponds to the Castelnuovo-Mumford regularity of the Cohen-Macaulay graded ring \(R/I_Z\).
For arbitrary fat points in \({\mathbb{P}}^n\), \textit{G. Fatabbi} and \textit{A. Lorenzini} [J. Pure Appl. Algebra 161, No. 1--2, 91--111 (2001; Zbl 0994.14028)] gave the following conjecture for the regularity index.
{ Conjecture 1}. For \(r = 1,\dots , n\) and for any linear \(r\)-subspace \(L\) of \({\mathbb{P}}^n\), set \(T(Z,L) := \lfloor\frac{ w_L(Z) + r - 2}{r}\rfloor\). Then \(\mathrm{reg}(Z) \leq \max\{T(Z, L) : L \subset{\mathbb{P}}^n\}\).
In Section 2 the authors prove that Conjecture 1 holds for schemes with \(n+3\) arbitrary fat points of \({\mathbb{P}}^n\).
In Section 3, Theorem 3.6 improves Segre's bound \(\mathrm{reg}(Z)\leq \max\{m_1+m_2-1,\lfloor\frac{ w_L(Z) + n - 2}{n}\rfloor\) for fat points \(Z\) in general position in \({\mathbb{P}}^n\).
In Section 4, the authors pose a modification of the Segre conjecture for the regularity index of a scheme of fat points, \(\mathrm{reg}(Z)\), and prove it holds for \(n = 3\). Segre's bound; generalised Segre's bound; regularity index; fat points Ballico, E.; Dumitrescu, O.; Postinghel, E.: On Segre's bound for fat points in pn. J. pure appl. Algebra 220, 2307-2323 (2016) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Divisors, linear systems, invertible sheaves On Segre's bound for fat points in \(\mathbb{P}^n\) | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The author describes three related techniques to work efficiently with power series in the context of (embedded) resolution of singularities, where the key problem is the construction of local invariants for singularities and their control under blowup and localization. The techniques are intended to provide a general framework for producing such invariants and observing their transformation rules.
1. One considers local flags in a regular ambient space, and local coordinates subordinate to the flag. The author constructs canonically new local flags on any blowup with centre transversal to the given flag. Choosing local coordinates subordinate to the flags before and after the blowup allows one to define significant invariants and to read off easily their change under the blowup.
2. The analog of the classical Gauss decomposition is proved for the group \(G\) of automorphisms of the formal (or convergent) power series in the variables \(x_1,\dots,x_n\) (i.e. local automorphisms of affine \(n\)-space): any \(\varphi \in G\) is a product \(\varphi =bus\), where \(b\) is an upper triangular automorphism, \(u\) is a lower unipotent de Joncquière automorphism and \(s\) is a permutation. As the \` Borel automorphisms\'\ \(b\) are just those which stabilize the flag with subordinate coordinates \(x_1,\dots,x_n\), one can restrict to automorphisms \(us\) when constructing subordinate coordinates out of ordinary ones. This is shown to be crucial for controlling a flag invariant under blowup or localization.
3. Initial ideals of an ideal of power series (with respect to monomial orders on \(\mathbb N^n\)) are a frequently used tool to investigate singularities, in particular the coordinate independent \textsl{generic initial ideal}. Here the author proposes to order the set of initial ideals with respect to all coordinate choices of a given ideal (by the ordering on monomial ideals through the lexicographic order taken on their minimal monomial generator systems). Then the minimal initial ideal is just the generic one, but the maximal initial ideal contains much more information on the singularity. Its existence is however not obvious and is shown in the paper. The Gauss decomposition of \(G\) is then used to determine coordinates realizing this maximal ideal.
The author summarizes the most important resolution invariants in the literature and indicates that they are special cases or variants of his flag invariants or minimal/maximal initial ideal. It is also interesting that the constructions in the paper are characteristic independent, thereby providing a potential contribution to the still unsolved characteristic \(p\) case. resolution of singularities; flag; Gauss decomposition of formal automorphisms; initial ideal Herwig Hauser, Three power series techniques, Proc. London Math. Soc. (3) 89 (2004), no. 1, 1 -- 24. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Local complex singularities, Invariants of analytic local rings, Modifications; resolution of singularities (complex-analytic aspects) Three power series techniques. | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system This article is devoted to an application of algebraical geometry to system theory.
Let s be an affine coordinate on the projective line \({\mathbb{P}}^ 1\), p an f-linear operator. The author states that the linear equations s p(z(s))\(=f(z(s))\) and algebraic coherent sheaves on \({\mathbb{P}}^ 1\), which are generated by global sections and are regular at the infinite point, are equivalent objects. From this result a generalization of the Martin- Hermann theorem and the Brunovsky theorem follows. linear system; projective line; linear equations; algebraic coherent sheaves; equivalent objects; Martin-Hermann theorem; Brunovsky theorem Lomadze, V.: Linear systems and coherent sheaves. Bull. acad. Sci. georgian SSR 126, N 1 (1987) Categories and theories, Algebraic methods, Elementary questions in algebraic geometry, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) Linear systems and coherent sheaves | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The first goal of this paper is to generalize in local complex geometry the main idea in \textit{W. L. Chow} and \textit{B. L. van der Waerden}' construction [Math. Ann. 113, 692-704 (1937; Zbl 0016.04004)]: associating the ``incidence divisor'' of a cycle with a given analytic family of cycles (the linear projective varieties of suitable dimension in the case of \(\mathbb{P}_N (\mathbb{C}))\).
Under very general hypotheses we associate a Cartier divisor on the parameter space in which arbitrary singularities are allowed to a local complete intersection.
The second goal is to study the singularities of functions obtained by integration of meromorphic cohomology classes on an analytic family of cycles. Our result enables us to control the pole order of the meromorphic functions on the parameter space from the pole order of the given cohomology class. local complex geometry; Cartier divisor; local complete intersection; singularities of functions; integration of meromorphic cohomology classes D. Barlet - J. Magnusson , Integration de classes de cohomologie méromorphes et diviseur d'incidence , Ann. Sci. École Norm. Sup. ( 4 ) 31 ( 1998 ), 811 - 842 . Numdam | MR 1664218 | Zbl 0963.32019 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Complete intersections Integration of meromorphic cohomology classes and incidence divisors | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(k\) be a perfect field of characteristic \(p>0\) and let \(X\) be a smooth separated scheme over \(k\) of dimension \(d\). Let \(D\) be a divisor with simple normal crossings, \(U=X-D\) its complement and \(D_1,\dots, D_h\) its irreducible components. Let \(\Lambda\) be a local ring over \(\mathbb{Z}[1/p,\zeta_p]\) and let \(\mathcal F\) be a locally constant constructible sheaf of free \(\Lambda\)-modules of finite rank on \(U\). Assume that the ramification of \(\mathcal F\) is non-degenerate along \(D\), meaning that the ramification along \(D\) is uniformly controlled at the generic points of the irreducible components and it is satisfied after shrinking \(X\) to a neighborhood of each generic point. Then one can define the characteristic cycle \(\mathrm{Char}(\mathcal F)\) of \(\mathcal F\) as an effective \(d\)-cycle with rational coefficients on the cotangent bundle \(T^*X\).
Under these hypotheses, the author shows that the characteristic cycle has coefficients in \(\mathbb{Z}[1/p]\). At the generic point of each irreducible component of the divisor at the boundary, the characteristic cycle of a sheaf is defined by some differential forms. For a morphism \(f: X' \to X '\) between smooth schemes over \(k\), of dimension \(d\) and \(d'\), with \(U =X -D\) and \(U' =X' -D'\), such that \(f^{-1}(U) =U'\), one defines \(f\) to be non-characteristic with respect to \(\mathcal F\) if \(f\) is non-characteristic with respect to \(\mathrm{Char}(\mathcal F) \in Z_d(T^*(X)_{\mathbb{Q}}\). The construction of the characteristic cycles commutes with pull-backs by non-characteristic morphisms, a result that allows one to deduce a characterization of the support of the characteristic cycle in terms of the restrictions to curves transversally meeting the boundary.
The results of this paper are non-logarithmic variants of the logarithmic version considered by the same author in [J. Inst. Math. Jussieu 8, No. 4, 769--829 (2009; Zbl 1177.14044)]. Saito, T.: Wild ramification and the cotangent bundle. J. Algebr. Geom. \textbf{26}(2017), 399-473 (2016) Étale and other Grothendieck topologies and (co)homologies, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Wild ramification and the cotangent bundle | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(K\) be a number field and \(A\) an abelian variety defined over \(K.\) The author studies the local-global principle for linear dependence of points for abelian varieties with \({\mathrm{End}}_{\bar K}(A)=\mathbb Z.\) The linear dependence of points \(P_1,\dots , P_n \in A(K)\) or \(A(k_v)\) means that \(a_{1}P_1+\dots +a_nP_n=0\) for rational integers \(a_1,\dots , a_n\) such that \(\gcd (a_1,\dots , a_n)\) divides the order of the torsion subgroup of \(A(K).\) The main result od the paper is that for a finite set of points \(S\) the equivalence of the following statements:
\begin{enumerate}
\item \(S\) is linearly dependent
\item For almost all primes \(v\) the set of images of elements of \(S\) via the reduction map \(r_v: A(K)\rightarrow A(k_v)\) is linearly dependent in \(A(k_v)\)
\end{enumerate}
holds iff \({\mathrm{rank}} A\leq 2\dim A.\) The corresponding result for elliptic curves is proven without the assumption on the endomorphism ring. Mordell-Weil groups; rank; linear dependence; local-global principle Rational points, Abelian varieties of dimension \(> 1\), Elliptic curves Hasse principle for linear dependence in Mordell-Weil groups | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(A\) be an integer-valued \(d \times n\) matrix of rank \(d\) and \(\beta \in \mathbb C^d\). The \(A\)-hypergeometric system is a \(D\)-module defined by \(A\) and \(\beta\). Here \(D = \mathbb C[x_1, \dots, x_n] \langle \partial_1, \dots, \partial_n \rangle\) is the Weyl algebra. In the present paper, the authors studies the irregularity of \(M_A(\beta)\). To do this, they introduce a filtration \(L = \{L_k D\}_k\) of \(D\) and use the associated graded ring \(\text{gr}^L D\). Let \((l_1, \dots, l_{2n}) \in \mathbb Q^{2n}\) such that \(l_1 + l_{n+1}\), \dots, \(l_n + l_{2n} \geq 0\). Then let \(L_k D\) be the vector space spanned by monomials \(x_1^{e_1} \cdots x_n^{e_n} \partial_1^{f_1} \cdots \partial_n^{f_n}\) such that \(\sum e_i l_i + \sum f_i e_{i+n} \leq k\). For such a filtration, they define an \((A, L)\)-umbrella, which is a cell complex, and the characteristic variety \(\text{Ch}^L(M_A(\beta))\), which is a subvariety of \(\text{Spec} \text{gr}^L D\). Furthermore, they define slopes of \(M_A(\beta)\) by using a family of filtrations parameterized by rational numbers. For an irreducible subvariety \(C \subset \text{Spec} \text{gr}^L D\), let \(\mu^{L,C}_{A,0}(\beta)\) be the multiplicity of \(\text{gr}^L M_A(\beta)\). The main result of this paper is the computation of \(\mu^{L,C}_{A,0}(\beta)\). As its consequence, they prove the converse of Hotta's theorem. That is, any \(A\)-hypergeometric system is homogeneous with respect to some filtration.
The authors, moreover, define the Euler-Koszul \(L\)-characteristic along \(C\): \(\mu_{A,i}^{L,C}(-; \beta)\) (\(i = 0\), \(1\), \dots), which is a generalized notion of \(\mu_{A,0}^{L, C}(\beta)\). They also compute it. \((A, L)\)-umbrella; irregularity sheaf; Euler-Koszul characteristic M. Schulze and U. Walther, Irregularity of hypergeometric systems via slopes along coordinate subspaces , preprint,\arxivmath/0608668v2[math.AG] Commutative rings of differential operators and their modules, Toric varieties, Newton polyhedra, Okounkov bodies, Filtered associative rings; filtrational and graded techniques Irregularity of hypergeometric systems via slopes along coordinate subspaces | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(P_1,\dots,P_s\) be \(s\) distinct points in the projective space \({\mathbb P}^n:={\mathbb P}_k^n\) with \(k\) an algebraically closed field of arbitrary characteristic and let \(m_1,\dots,m_s\) be positive integers. The zero-dimensional scheme \(Z:=m_1P_1+\dots+m_sP_s\) defined by all forms in the polynomial ring \(R:=k[x_0,\dots,x_n]\) vanishing at \(P_i\) with multiplicity at least \(m_i\) is called a set of fat points in \({\mathbb P}^n\) and \(I:=I_Z\) is its defining ideal.
Let consider the graded ring \(A := R/I\), which is the homogeneous coordinate ring of \(Z\). It is well known that \(A =\sum_{t\geq 0}A_ t\) is a one-dimensional Cohen-Macaulay graded ring whose multiplicity is \(e(A) = \sum_{i=1}^{s}{{m_i + n-1}\choose{ n}}\).
Furthermore, the Hilbert function \(H_A(t) := \dim_k A_t\) strictly increases until it reaches the multiplicity, at which it stabilizes. The regularity index of \(A\) (or of the fat points \(Z\)) is defined to be the least integer \(t\) such that \(H_A(t) = e(A)\), and we will denote it by \(\mathrm{reg}(Z)\) (or by \(\mathrm{reg}(A)\)).
In this paper, the author shows a formula to compute the regularity index of \(s+2\) fat points not lying on a linear \((s-1)\)-space in \({\mathbb P}^n\), \(s\leq n\) (see Theorem 3.4). This result shows that the Segre bound is attained in the case of \(s+2\) fat points not lying on a linear \((s-1)\)-space. Since \(s+2\) fat points in general position never lie on a linear \((s-1)\)-space, the result also generalizes a formula to compute the regularity index of fat points in general position in \({\mathbb P}^n\) given by \textit{M. V. Catalisano, N. V. Trung} and \textit{G. Valla} [Proc. Am. Math. Soc. 118, No. 3, 717--724 (1993; Zbl 0787.14030)]. fat point; regularity index; zero-scheme DOI: 10.1080/00927872.2011.593385 Divisors, linear systems, invertible sheaves, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Regularity index of \(s + 2\) fat points not on a linear \((s - 1)\)-space | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The subject of this work are linear systems in \({\mathbb P}^n\) of type \({\mathcal L} = {\mathcal L}_{n,d}(-\sum _{i=1}^h m_iP_i)\), \(P_i\in {\mathbb P}^n\), \(m_i\in {\mathbb N}\); which are given by hypersurfaces of degree \(d\) passing through \(h\) generic points \(P_i\) with multiplicities \(\geq m_i\).
Even for \(n=2\), the dimension of \({\mathcal L}\) is not known in general, but several equivalent conjectures (the first due to B. Segre) state that the dimension is the expected one except in the cases when the linear system contains a multiple fixed component which is a \((-1)\)-curve. In a previous paper, the author proposed yet other two equivalent forms for such conjectures, stating that \({\mathcal L}_{2,d}(-\sum _{i=1}^h m_iP_i)\) is special (i.e. it does not have the expected dimension), if and only if it is ``numerically special'' or ``cohomologically special'', where the main interest for those two concepts is that they could be generalized to \(n\geq 3\), where no general conjecture for the dimension of \({\mathcal L}\) is known.
We say that \({\mathcal L}\) is numerically special if it exists an \(\alpha\)-special effect variety \(Y\) for \({\mathcal L}\), i.e. an irreducible variety \(Y\) such that \({\mathcal L}-\alpha Y\) has positive virtual dimension greater than \({\mathcal L}\) (a few more conditions are required if \(\dim Y = n-1\)). We say that \({\mathcal L}\) is cohomologically special, instead, if it exists a \(h^1\)-special effect variety \(Y\), i.e. an irreducible \(Y\) such that \({\mathcal L}- Y\) has positive dimension, \(h^0({\mathcal L}|_Y)=0\) and \(h^1({\mathcal L}|_Y)>h^2({\mathcal L}-Y)\) (an example showing that the two definitions do not coincide is given).
The author conjectures that for \({\mathcal L}\) to be special is equivalent to being numerically special and also to being cohomologically special. In the case \(n=2\), \(\alpha\)-special effect varieties and \(h^1\)-special effect varieties are actually \((-1)\)-curves, while for \(n\geq 3\) examples are given using rational normal curves, hypersurfaces or linear spaces, and those covers most known examples of special \({\mathcal L}\)'s.
Moreover, the case of linear systems in a product \({\mathbb P}^{n_1}\times {\mathbb P}^{n_2}\times\dots\times {\mathbb P}^{n_t}\) is studied and examples (known and new) are given of systems whose speciality is due to special effect varieties. linear systems; multiple points; fat poins; Hilbert function Bocci, C.: Special effect varieties in higher dimension. Collect. Math. \textbf{56}(3), 299-326 (2005). ISSN: 0010-0757 Divisors, linear systems, invertible sheaves, Projective techniques in algebraic geometry, Singularities of curves, local rings Special effect varieties in higher dimension | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system In {\S}1 the author develops a general theory of divisors and linear systems on an integral projective curve with Gorenstein singularities. A (generalized) divisor is synonymous with a fractional ideal, i.e. a subsheaf of the constant sheaf of the function field of the curve C which is coherent as \({\mathcal O}_ C\)-module. The sum of two divisors is defined only if one of them is a Cartier divisor. The notion of complete linear systems, Riemann-Roch theorem and Serre duality theorem remain valid. As an example the author gives a new proof of a theorem of Fujita which says that the canonical system \(| K|\) has no base points if \(p_ a(C)>0\), and is very ample if \(p_ a(C)>1\). - In {\S}2 he applies the results of {\S}1 to prove a theorem about an irreducible plane curve of degree \(k.\) If Z is a subscheme of finite length d of C, it gives a precise upper bound for \(h^ 0({\mathcal O}_ C(Z))\) in terms of k and d and tells when the bound is attained. The proof is by induction on k, and even when C is non-singular (the case Max Noether considered) one has to deal with singular curves in the course of induction, and this is the point ignored by Noether. subscheme of irreducible plane curve; linear systems; integral projective curve with Gorenstein singularities; Cartier divisor R. Hartshorne, Generalized divisors on Gorenstein curves and a theorem of Noether, J. Math. Kyoto Univ. 26 (1986), 375--386. Divisors, linear systems, invertible sheaves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Special algebraic curves and curves of low genus Generalized divisors on Gorenstein curves and a theorem of Noether | 0 |
Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \((R,\mathfrak{m})\) be a Noetherian \(d\)-dimensional local ring such that
the nilradical \(N(\hat{R})\) of the \(\mathfrak{m}\)-adic completion \(\hat{R}\) of
\(R\) has dimension < \(d\). Let \(\mathcal{I}(1)=\{\mathcal{I}(1)_{i}
\},\ldots,\mathcal{I}(r)=\{\mathcal{I}(r)_{i}\}\) be a system of \(r\)
filtrations on \(R\) (not necessarily Noetherian i.e. \(\oplus_{n\geq
0}\mathcal{I}(j)_{n}\) may not be finitely generated \(R\)-algebras), where
\(\mathcal{I}(j)_{i}\) are \(\mathfrak{m}\)-primary ideals. The authors prove that
for a finitely generated \(R\)-module \(M\) the function
\[
P(n_{1},\ldots,n_{r}):=\lim_{m\rightarrow\infty}\frac{\lambda(M/\mathcal{I}
(1)_{mn_{1}}\cdots\mathcal{I}(r)_{mn_{r}}M)}{m^{d}}
\]
is defined and is a homogeneous polynomial of total degree \(d\) with real
coefficients. Appropriately normalized coefficients of \(P\) are, by definition,
mixed multiplicities of \(M\) with respect to the filtrations \(\mathcal{I}
(1),\ldots,\mathcal{I}(r).\) They show many of the classical properties of
mixed multiplicities known for \(\mathfrak{m}\)-primary ideals \(I_{1}
,\ldots,I_{r}\) (i.e. for filtrations \(\mathcal{I}(j)_{i}:=I_{j}^{i}\)), in
particular the Minkowski inequalities. local ring; multiplicity; filtration;mixed multiplicity; Minkowski inequality Multiplicity theory and related topics, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Mixed multiplicities of filtrations | 0 |
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