empirical processes and random projections in this note we establish some bounds on the supremum of certain empirical processes indexed by sets of functions with the same l2 norm we present several geometric applications of this result the most important of which is a sharpening of the johnson lindenstrauss embedding lemma our results apply to a large class of random matrices as we only require that the matrix entries have a subgaussian tail

linear free resolutions and minimal multiplicity let s k x x be a polynomial ring over a field and let a4 a m be a finitely generated graded module in the most interesting case a4 is an ideal of s for a given natural number p there is a great interest in the question can m be generated by homogeneous elements of degree p no simple answer say in terms of the local cohomology of m is known but somewhat surprisingly the stronger question can the jth syzygy of m be generated by elements of degree qp j for all j 0 l n does admit a simple response the following is neither new nor difficult to prove though it seems not well known

implicitization of rational surfaces using toric varieties abstract a parameterized surface can be represented as a projection from a certain toric surface this generalizes the classical homogeneous and bihomogeneous parameterizations we extend to the toric case two methods for computing the implicit equation of such a rational parameterized surface the first approach uses resultant matrices and gives an exact determinantal formula for the implicit equation if the parameterization has no base points in the case the base points are isolated local complete intersections we show that the implicit equation can still be recovered by computing any non zero maximal minor of this matrix the second method is the toric extension of the method of moving surfaces and involves finding linear and quadratic relations syzygies among the input polynomials when there are no base points we show that these can be put together into a square matrix whose determinant is the implicit equation its extension to the case where there are base points is also explored

on the yang lee edge singularity for ising model on nonhomogeneous structures abstract we studied distribution of zeros of the partition function of ferromagnetic ising model near the yang lee edge on two self similar structures we have shown that the nature of associated critical behavior crucially depends on the local lattice structure if the sites of higher coordination number make an infinite connected cluster then we find an usual power law behavior while a logarithmic divergence of the correlation length develops near the edge in the case that these sites make only finite islands we reveal also a close connection between yang lee edge critical behavior and critical behavior of a simple zero field gaussian model on the same structures

maximal cohen macaulay modules and gorenstein algebras abstract let b be a graded cohen macaulay quotient of a gorenstein ring r it is known that sections of the dual of the canonical module kb can be used to construct gorenstein quotients of r the purpose of this paper is to place this method of construction into a broader context if m is a maximal cohen macaulay b module whose sheafified top exterior power is a twist of kb and if m satisfies certain additional homological conditions then regular sections of m can again be used to construct gorenstein quotients of r on cohen macaulay quotients the normal module the first koszul homology module and several other associated modules all have sheafified top exterior power equal to a twist of kb if additional restrictions are placed on the cohen macaulay quotients then these modules will satisfy the required additional homological conditions this places the canonical module within a broad family of easily manipulated maximal cohen macaulay modules whose sections can be used to construct gorenstein quotients of r

gl v invariant ulrich bundles on grassmannians in this paper we give a full classification of all homogeneous ulrich bundles on a grassmannian mathrm gr k n of k planes on mathbb p n

evolutions symbolic squares and fitting ideals given a reduced local algebra t over a suitable ring or field k we study the question of whether there are nontrivial algebra surjections r t which induce isomorphisms r k t t k such maps called evolutions arise naturally in the study of hecke algebras as they implicitly do in the recent work of wiles taylor wiles and flach we show that the existence of non trivial evolutions of an algebra t can be characterized in terms of the symbolic square of an ideal defining t we give a characterization of the symbolic square in terms of fitting ideals using this and other techniques we show that certain classes of reduced algebras codimension 2 cohen macaulay codimension 3 gorenstein licci algebras in general and some others admit no nontrivial evolutions on the other hand we give examples showing that non trivial evolutions of reduced cohenmacaulay algebras of codimension 3 do exist in every positive characteristic

homogeneous acm bundles on a grassmannian abstract in this paper we characterize homogeneous arithmetically cohen macaulay acm for short bundles on grassmannians gr k n and we construct irreducible families of indecomposable acm bundles on a grassmannian gr k n of arbitrary high rank and dimension as a consequence we prove that all grassmann varieties gr k n are of wild representation type unless the projective spaces gr 0 n and gr n 1 n and the hyperquadric gr 1 3 in p 5 which are of finite representation type

acm sheaves of pure rank two on reducible hyperquadrics abstract we classify a special type of arithmetically cohen macaulay sheaves of rank two on reducible and reduced quadric hypersurfaces as a consequence we show that a reducible and reduced quadric surface is of wild type

ulrich modules over cyclic quotient surface singularities abstract in this paper we characterize ulrich modules over cyclic quotient surface singularities using the notion of special cohen macaulay modules we also investigate the number of indecomposable ulrich modules for a given cyclic quotient surface singularity and show that the number of exceptional curves in the minimal resolution determines a boundary on the number of indecomposable ulrich modules

vector bundles whose restriction to a linear section is ulrich an ulrich sheaf on an n dimensional projective variety x subseteq mathbb p n is an initialized acm sheaf which has the maximum possible number of global sections using a construction based on the representation theory of roby clifford algebras we prove that every normal acm variety admits a reflexive sheaf whose restriction to a general 1 dimensional linear section is ulrich we call such sheaves delta ulrich in the case n 2 where delta ulrich sheaves satisfy the property that their direct image under a general finite linear projection to mathbb p 2 is a semistable instanton bundle on mathbb p 2 we show that some high veronese embedding of x admits a delta ulrich sheaf with a global section

recipes for stable linear embeddings from hilbert spaces to mathbb r m we consider the problem of constructing a linear map from a hilbert space mathcal h possibly infinite dimensional to mathbb r m that satisfies a restricted isometry property rip on an arbitrary signal model i e a subset of mathcal h we present a generic framework that handles a large class of low dimensional subsets but also unstructured and structured linear maps we provide a simple recipe to prove that a random linear map satisfies a general rip with high probability we also describe a generic technique to construct linear maps that satisfy the rip finally we detail how to use our results in several examples which allow us to recover and extend many known compressive sampling results

rank two acm bundles on the del pezzo threefold with picard number 3 abstract a del pezzo threefold f with maximal picard number is isomorphic to p 1 p 1 p 1 in the present paper we completely classify locally free sheaves e of rank 2 such that h i f e t 0 for i 1 2 and t z such a classification extends similar results proved by e arrondo and l costa regarding del pezzo threefolds with picard number 1

complete description of determinantal representations of smooth irreducible curves abstract determinantal representations of algebraic curves are interesting in themselves and their classification is equivalent to the simultaneous classification of triples of matrices we present a complete description of determinantal representations of smooth irreducible curves over any algebraically closed field we used the notion of the class of divisors of the vector bundle corresponding to a determinantal representation we prove that two determinantal representations of a smooth curve f are equivalent if and only if the classes of divisors of the corresponding vector bundles coincide we give a precise characterization of those classes of divisors that arise from vector bundles corresponding to determinantal representations of f then we obtain a parametrization of determinantal representations of f up to equivalence by the points of the jacobian variety of f not on some exceptional subvariety in particular it follows that any smooth curve of order 3 or greater possesses an infinite number of nonequivalent determinantal representations we also specialize our results to symmetrical and self adjoint representations

macaulay style formulas for sparse resultants we present formulas for computing the resultant of sparse polynomials as a quotient of two determinants the denominator being a minor of the numerator these formulas extend the original formulation given by macaulay for homogeneous polynomials

the geometry of ulrich bundles on del pezzo surfaces abstract given a smooth del pezzo surface x d p d of degree d we isolate the essential geometric obstruction to a vector bundle on x d being an ulrich bundle by showing that an irreducible curve d of degree dr on x d represents the first chern class of a rank r ulrich bundle on x d if and only if the kernel bundle of the general smooth element of d admits a generalized theta divisor moreover we show that any smooth arithmetically gorenstein surface whose ulrich bundles admit such a characterization is necessarily del pezzo this result is applied to produce new examples of complete intersection curves with semistable kernel bundle and also combined with work of farkas musta and popa to relate the existence of ulrich bundles on x d to the minimal resolution conjecture for curves lying on x d in particular we show that a smooth irreducible curve d of degree 3r lying on a smooth cubic surface x 3 represents the first chern class of an ulrich bundle on x 3 if and only if the minimal resolution conjecture holds for the general smooth element of d

the representation type of segre varieties abstract the goal of this work is to prove that all segre varieties n 1 n s p n n i 1 s n i 1 1 unless the quadric surface in p 3 support families of arbitrarily large dimension and rank of simple ulrich and hence acm vector bundles therefore they are of wild representation type

gorenstein biliaison and acm sheaves abstract let x be a normal arithmetically gorenstein scheme in p n we give a criterion for all codimension two acm subschemes of x to be in the same gorenstein biliaison class on x in terms of the category of acm sheaves on x these are sheaves that correspond to the graded maximal cohen macaulay modules on the homogeneous coordinate ring of x using known results on mcm modules we are able to determine the gorenstein biliaison classes of codimension two subschemes of certain varieties including the nonsingular quadric surface in p 3 and the cone over it in p 4 as an application we obtain a new proof of some theorems of lesperance about curves in p 4 and answer some questions he raised

surfaces of minimal degree of tame representation type and mutations of cohen macaulay modules we provide two examples of smooth projective surfaces of tame cm type by showing that the parameter space of isomorphism classes of indecomposable acm bundles with fixed rank and determinant on a rational quartic scroll in p5p5 is either a single point or a projective line these turn out to be the only smooth projective acm varieties of tame cm type besides elliptic curves 1 for surfaces of minimal degree and wild cm type we classify rigid ulrich bundles as fibonacci extensions for f0f0 and f1f1 embedded as quintic or sextic scrolls a complete classification of rigid acm bundles is given

liaison classes of modules abstract we propose a concept of module liaison that extends gorenstein liaison of ideals and provides an equivalence relation among unmixed modules over a commutative gorenstein ring analyzing the resulting equivalence classes we show that several results known for gorenstein liaison are still true in the more general case of module liaison in particular we construct two maps from the set of even liaison classes of modules of fixed codimension into stable equivalence classes of certain reflexive modules as a consequence we show that the intermediate cohomology modules and properties like being perfect cohen macaulay buchsbaum or surjective buchsbaum are preserved in even module liaison classes furthermore we prove that the module liaison class of a complete intersection of codimension one consists of precisely all perfect modules of codimension one

two observations regarding embedding subsets of euclidean spaces in normed spaces abstract this paper contains two results concerning linear embeddings of subsets of euclidean space in low dimensional normed spaces the first is an improvement of the known dependence on in dvoretzky s theorem from order of 2 to order of except for log factors the second is a joint generalization of milman s version of dvoretzky s theorem and a recent generalization by klartag and mendelson of the johnson lindenstrauss lemma

ulrich bundles on the degree six segre fourfold abstract we completely characterize the bigraded resolutions of ulrich bundles of arbitrary rank on the segre fourfold p 2 p 2 we characterize the ulrich bundles v of arbitrary rank on p 2 p 2 with h 1 v 0 or with h 1 v 1 1 0 or obtained as pullback from p 2 and we construct more complicated examples

a note on the multiplicity of determinantal ideals abstract herzog huneke and srinivasan have conjectured that for any homogeneous k algebra the multiplicity is bounded above by a function of the maximal degrees of the syzygies and below by a function of the minimal degrees of the syzygies the goal of this paper is to establish the multiplicity conjecture of herzog huneke and srinivasan about the multiplicity of graded cohen macaulay algebras over a field k for k algebras k x 1 x n i when i is a determinantal ideal of arbitrary codimension

zipping tate resolutions and exterior coalgebras abstract we conjecture what the cone of hypercohomology tables of bounded complexes of coherent sheaves on projective spaces is when we have specified regularity conditions on the cohomology sheaves of this complex and its dual there is an injection from this cone into the cone of homological data sets of squarefree modules over a polynomial ring k x 1 x n and we conjecture that this is an isomorphism the tate resolutions of a complex of coherent sheaves on projective space p w and the exterior coalgebra on x 1 x n may be amalgamated together to form a complex of free sym i x i w modules a procedure introduced by cox and materov via a reduction i x i w i x i k we get a complex of free modules over k x 1 x n the extremal rays in the cone of squarefree complexes are conjecturally given by triplets of pure free squarefree complexes introduced in 15 we describe the corresponding classes of hypercohomology tables a class which generalizes vector bundles with supernatural cohomology we also show how various pure resolutions in the literature like resolutions of modules supported on determinantal varieties and tensor complexes may be obtained by the first part of the procedure

majorizing measures without measures we give a reformulation of majorizing measures that does not involve measures but rather special sequences of partitions this formulation is more convenient to perform chaining with different distances

metric and classical fidelity uncertainty relations for random unitary matrices we analyze uncertainty relations on finite dimensional hilbert spaces expressed in terms of classical fidelity which are stronger than metric uncertainty relations introduced by fawzi hayden and sen we establish the validity of fidelity uncertainty relations for random unitary matrices with optimal parameters up to universal constants which improves upon known results for the weaker notion of metric uncertainty this result is then applied to locking classical information in quantum states and allows to obtain optimal locking in hellinger distance improving upon previous results on locking in the total variation distance both by strengthening the metric used and by improving the dependence on parameters we also show that general probabilistic estimates behind the main theorem can be used to prove existence of data hiding schemes with bayesian type guarantees as a byproduct of our approach we obtain existence of almost euclidean subspaces of the matrix spaces with a better dimension distortion dependence than allowed in previously known constructions

regularity of gaussian processes on donne la validite de la conjecture de fernique 1974 sur l existence de mesures de majoration pouvant caracteriser la limitation des processus gaussiens

on a result of gelfand kapranov and zelevinsky abstract in this paper we give new elementary proofs of basic results due to gelfand kapranov and zelevinsky which express discriminants and results in terms of determinants of direct images of stably twisted cayley koszul complexes of sheaves

hilbert functions of irreducible arithmetically gorenstein schemes abstract in this paper we compute the hilbert functions of irreducible or smooth and reduced arithmetically gorenstein schemes that are twisted anti canonical divisors on arithmetically cohen macaulay schemes we also prove some folklore results characterizing the hilbert functions of irreducible standard determinantal schemes and we use them to produce a new class of functions that occur as hilbert functions of irreducible or smooth and reduced arithmetically gorenstein schemes in any codimension

families of artinian and one dimensional algebras abstract the purpose of this paper is to study families of artinian or one dimensional quotients of a polynomial ring r with a special look to level algebras let gradalg h r be the scheme parametrizing graded quotients of r with hilbert function h let b a be any graded surjection of quotients of r with hilbert function h b 1 h 1 h j and h a respectively if dim a 0 respectively dim a depth a 1 and a is a truncation of b in the sense that h a 1 h 1 h j 1 0 0 respectively h a 1 h 1 h j 1 for some h j then we show there is a close relationship between gradalg h a r and gradalg h b r concerning e g smoothness and dimension at the points a and b respectively provided b is a complete intersection or provided the castelnuovo mumford regularity of a is at least 3 sometimes 2 larger than the regularity of b in the complete intersection case we generalize this relationship to non truncated artinian algebras a which are compressed or close to being compressed for more general artinian algebras we describe the dual of the tangent and obstruction space of graded deformations in a manageable form which we make rather explicit for level algebras of cohen macaulay type 2 this description and a linkage theorem for families allow us to prove a conjecture of iarrobino on the existence of at least two irreducible components of gradalg h r h 1 3 6 10 14 10 6 2 whose general elements are artinian level algebras of type 2

deformations of modules of maximal grade and the hilbert scheme at determinantal schemes let r be a polynomial ring and m a finitely generated graded r module of maximal grade which means that the ideal it a generated by the maximal minors of a homogeneous presentation matrix a of m has maximal codimension in r suppose x proj r it a is smooth in a sufficiently large open subset and dimx 1 then we prove that the local graded deformation functor of m is isomorphic to the local hilbert scheme functor at x proj r under a weak assumption which holds if dimx 2 under this assumption we get that the hilbert scheme is smooth at x and we give an explicit formula for the dimension of its local ring as a corollary we prove a conjecture of r m miro roig and the author that the closure of the locus of standard determinantal schemes with fixed degrees of the entries in a presentation matrix is a generically smooth component v of the hilbert scheme also their conjecture on the dimension of v is proved for dimx 1 the cohomology h i nx of the normal sheaf of x in proj r is shown to vanish for 1 i dimx 2 finally the mentioned results slightly adapted remain true replacing r by any cohen macaulay quotient of a polynomial ring

on the cohen macaulay modules of graded subrings we give several characterizations for the linearity property for a maximal cohen macaulay module over a local or graded ring as well as proofs of existence in some new cases in particular we prove that the existence of such modules is preserved when taking segre products as well as when passing to veronese subrings in low dimensions the former result even yields new results on the existence of finitely generated maximal cohen macaulay modules over non cohen macaulay rings

dimension of families of determinantal schemes a scheme x p n c of codimension c is called standard determinantal if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous t x t c 1 matrix and x is said to be good determinantal if it is standard determinantal and a generic complete intersection given integers a 0 a 1 a t c 2 and b 1 b t we denote by w b a c hilb p p n c resp w s b a the locus of good resp standard determinantal schemes x c p n c of codimension c defined by the maximal minors of a t x t c 1 matrix f ij i 1 t j 0 t c 2 where f ij k x 0 x 1 x n c is a homogeneous polynomial of degree a j b i in this paper we address the following three fundamental problems to determine 1 the dimension of w b a resp w s b a in terms of a j and b i 2 whether the closure of w b a is an irreducible component of hilb p p n c and 3 when hilb p p n c is generically smooth along w b a concerning question 1 we give an upper bound for the dimension of w b a resp w s b a which works for all integers a 0 a 1 a t c 2 and b 1 b t and we conjecture that this bound is sharp the conjecture is proved for 2 2 and for c 5 under certain numerical assumptions

linkage and the koszul homology of ideals 0 introduction let r be a local ring and i an ideal of r associated to i are several graded algebras the symmetric algebra of the module i sym i the rees algebra of i defined to be the algebra r i gn o inf and the associated graded algebra of i gr1 r rii g ii 2 in in 1 i i i our purpose in this paper is to study the behavior of these graded algebras when we change i through linkage recall the definition of linkage found in the paper of peskine and szpiro 22

ulrich sheaves and higher rank brill noether theory abstract we show that the existence of an ulrich sheaf on a projective variety x p n is equivalent to the solution of a possibly higher rank brill noether problem for a curve on x that is rarely general in moduli in addition we exhibit a large family of curves for which this brill noether problem admits a solution and we show that existence of an ulrich sheaf for a finite morphism of smooth projective varieties of any dimension implies sharp numerical constraints involving the degree of the map and the ramification divisor

minimal resolutions chow forms and ulrich bundles on k3 surfaces the minimal resolution conjecture mrc for points on a projective varietyx p predicts that the minimal graded free resolution of a general set x of points is as simple as the geometry of x allows originally the most studied case has been that when x p see epsw the general form of the mrc for subvarieties x p was formulated in mus and fmp the betti diagram of a large enough set x consisting of general points is obtained from the betti diagram of x by adding two rows indexed by u 1 and u where u is an integer depending on all differences bi 1 u 1 bi u are known and depend on the hilbert polynomial px and i u and see fmp the minimal resolution conjecture for general points on x predicts that

non commutative desingularization of determinantal varieties i we show that determinantal varieties defined by maximal minors of a generic matrix have a non commutative desingularization in that we construct a maximal cohen macaulay module over such a variety whose endomorphism ring is cohen macaulay and has finite global dimension in the case of the determinant of a square matrix this gives a non commutative crepant resolution

majorizing measures the generic chaining majorizing measures provide bounds for the supremum of stochastic processes they represent the most general possible form of the chaining argument going back to kolmogorov majorizing measures arose from the theory of gaussian processes but they now have applications far beyond this setting the fundamental question is the construction of these measures this paper focuses on the tools that have been developed for this purpose and in particular the use of geometric ideas applications are given to several natural problems where entropy methods are powerless

on the normal sheaf of determinantal varieties o roig abstract let x be a standard determinantal scheme xp n of codimension c i e a scheme defined by the maximal minors of a t t c 1 homogeneous polynomial matrix a in this paper we study the main features of its normal sheaf nx we prove that under some mild restrictions 1 there exists a line bundle l on x sing x such that nx l is arithmetically cohen macaulay and even more it is ulrich whenever the entries of a are linear forms 2 nx is simple hence indecomposable and finally 3 nx is semi stable provided the entries of a are linear forms

difference chow form in this paper the generic intersection theory for difference varieties is presented precisely the intersection of an irreducible difference variety of dimensiond 0 and order h with a generic difference hypersurface of orders is shown to be an irreducible difference variety of dimensiond 1 and orderh s based on the intersection theory the difference chow form for an irreducible difference variety is defined furthermore it is shown that the difference chow form of an irreducible difference variety v is transformally homogenous and has the same order as v

the yang lee edge singularity for the ising model on two sierpinski fractal lattices we study the distribution of zeros of the partition function of the ferromagnetic ising model near the yang lee edge on two sierpiski type lattices we have shown that relevant correlation length displays a logarithmic divergence near the edge where is a constant and h distance from the edge in the case of a modified sierpinski gasket with a nonuniform coordination number it is demonstrated that this critical behavior can be related to the critical behavior of a simple zero field gaussian model of the same structure we have shown that there is no such connection between these two models on a second lattice that has a uniform coordination number these findings suggest that fluctuations of the lattice coordination number of a nonhomogeneous self similar structure exert the crucial influence on the critical behavior of both models

lp moments of random vectors via majorizing measures abstract for a random vector x in r n we obtain bounds on the size of a sample for which the empirical pth moments of linear functionals are close to the exact ones uniformly on a convex body k r n we prove an estimate for a general random vector and apply it to several problems arising in geometric functional analysis in particular we find a short lewis type decomposition for any finite dimensional subspace of l p we also prove that for an isotropic log concave random vector we only need n p 2 log n sample points so that the empirical pth moments of the linear functionals are almost isometrically the same as the exact ones we obtain a concentration estimate for the empirical moments the main ingredient of the proof is the construction of an appropriate majorizing measure to bound a certain gaussian process

ulrich schur bundles on flag varieties abstract in this paper we study equivariant vector bundles on partial flag varieties arising from schur functors we show that a partial flag variety with three or more steps does not admit an ulrich bundle of this form with respect to the minimal ample class we classify ulrich bundles of this form on two step flag varieties f 1 n 1 n f 2 n 1 n f 2 n 2 n f k k 1 n and f k k 2 n we give a conjectural description of the two step flag varieties which admit such ulrich bundles

relative beilinson monad and direct image for families of coherent sheaves the higher direct image complex of a coherent sheaf or finite complex of coherent sheaves under a projective morphism is a fundamental construction that can be defined via a cech complex or an injective resolution both inherently infinite constructions using exterior algebras and relative versions of theorems of beilinson and bernstein gel fand gel fand we give an alternate description in finite terms using this description we can characterize the generic complex over the variety of finite free complexes of a given shape as the direct image of an easily described vector bundle we can also give explicit descriptions of the loci in the base spaces of flat families of sheaves in which some cohomological conditions are satisfied for example the loci where vector bundles on projective space split in a certain way or the loci where a projective morphism has higher dimensional fibers our approach is so explicit that it yields an algorithm suited for computer algebra systems

intersection theory in differential algebraic geometry generic intersections and the differential chow form in this paper an intersection theory for generic differential polynomials is presented the intersection of an irreducible differential variety of dimension d and order h with a generic differential hypersurface of order s is shown to be an irreducible variety of dimension d 1 and order h s as a consequence the dimension conjecture for generic differential polynomials is proved based on intersection theory the chow form for an irreducible differential variety is defined and most of the properties of the chow form in the algebraic case are established for its differential counterpart furthermore the generalized differential chow form is defined and its properties are proved as an application of the generalized differential chow form the differential resultant of n 1 generic differential polynomials in n variables is defined and properties similar to that of the macaulay resultant for multivariate polynomials are proved

a remark on ulrich and acm bundles abstract i show that on any smooth projective ordinary curve of genus at least two and a projective embedding there is a natural example of a stable ulrich bundle for this embedding namely the sheaf b x 1 of locally exact differentials twisted by o x 1 given by this embedding and in particular there exist ordinary varieties of any dimension which carry ulrich bundles in higher dimensions assuming x is frobenius split variety i show that b x 1 is an acm bundle and if x is also a calabi yau variety and p 2 then b x 1 is not a direct sum of line bundles in particular i show that b x 1 is an acm bundle on any ordinary calabi yau variety i also prove a characterization of projective varieties with trivial canonical bundle such that b x 1 is acm for some projective embedding datum all such varieties are frobenius split with trivial canonical bundle

chow form for projective differential variety abstract in this paper a generic intersection theorem in projective differential algebraic geometry is given then the chow form for an irreducible projective differential variety is defined and basic properties of the differential chow form in affine differential case are shown to be valid for its projective differential counterpart finally we apply the differential chow form to a result of linear dependence over projective varieties given by kolchin

resultants and chow forms via exterior syzygies let w be a vector space of dimension n 1 over a field k the chow divisor of a k dimensional variety x in p p w is the hypersurface in the grassmannian gk 1 of planes of codimension k 1 in p whose points over the algebraic closure of k are the planes that meet x the chow form of x is the defining equation of the chow divisor for example the resultant of k 1 forms of degree e in k 1 variables is the chow form of p embedded by the e th veronese mapping in p with n k e k 1 more generally the chow divisor of a k cycle i ni vi on projective space is defined to be i nidi where di is the chow divisor of vi the chow divisor of a sheaf f with k dimensional support is the chow divisor of the associated k cycle of f in this paper we will give a new expression for the chow divisor and apply it to give explicit formulas in many new cases starting with a sheaf f on p we use exterior algebra methods to define a canonical and effectively computable chow complex of f on each grassmannian of planes in p if f has k dimensional support we show that the chow form of f is the determinant of the chow complex of f on the grassmannian of planes of codimension k 1 the beilinson monad of f beilinson 1978 is the chow complex of f on the grassmannian of 0 planes that is on p itself in particular we are able to give explicit determinantal and pfaffian formulas for resultants in some cases where no polynomial formulas were known for example the horrocks mumford bundle gives rise to polynomial formulas for the resultant of five homogeneous forms of degrees 4 6 or 8 in five variables the easiest of our new formulas to write down is for the resultant of 3 quadratic forms in three variables the chow form of the veronese surface in p using the tangent bundle of p conclude that it can be written in bezout form described below as the pfaffian of the matrix 0 245 345 135 045 035 145 235

oscillatory behavior of critical amplitudes of the gaussian model on a hierarchical structure we studied oscillatory behavior of critical amplitudes for the gaussian model on a hierarchical structure presented by a modified sierpinski gasket lattice this model is known to display nonstandard critical behavior on the lattice under study the leading singular behavior of the correlation length ensuremath xi near the critical coupling k k c is modulated by a function which is periodic in mathrm ln mathrm ln k c ensuremath k we have also shown that the common finite size scaling hypothesis according to which for a finite system at criticality ensuremath xi should be of the order of the size of the system is not applicable in this case as a consequence of this the exact form of the leading singular behavior of ensuremath xi differs from the one described earlier which was based on the finite size scaling assumption

special ulrich bundles on regular weierstrass fibrations the main goal of this short paper is to prove the existence of rank 2 simple and special ulrich bundles on a wide class of elliptic surfaces namely on regular weierstrass fibrations pi s rightarrow mathbb p 1 alongside we also show the existence of rank 2 weakly ulrich sheaves on arbitrary weierstrass fibrations s rightarrow c 0 and we deal with the non existence of rank one ulrich bundles on them

sheaves with canonical determinant on cohen macaulay schemes abstract this paper studies the class of sheaves which lie on arithmetically cohen macaulay schemes and which have as determinant a twist of the canonical sheaf special emphasis is placed on finding minimal sufficient cohomological conditions which ensure that the top dimensional component of regular sections of the dual of these sheaves vanish along arithmetically gorenstein schemes duals of odd rank buchsbaum rim sheaves normal sheaves to licci schemes certain sheafified koszul homology modules and various other families are shown to satisfy the requisite conditions

sheaf cohomology and free resolutions over exterior algebras in this paper we derive an explicit version of the bernstein gel fand gel fand bgg correspondence between bounded complexes of coherent sheaves on projective space and minimal doubly infinite free reso lutions over its koszul dual exterior algebra among the facts about the bgg correspondence that we derive is that taking homology of a complex of sheaves corresponds to taking the linear part of a resolution over the exterior algebra we explore the structure of free resolutions over an exterior algebra for example we show that such resolutions are eventually dominated by their linear parts in the sense that erasing all terms of degree 1 in the complex yields a new complex which is eventually exact as applications we give a construction of the beilinson monad which expresses a sheaf on projective space in terms of its cohomology by using sheaves of differential forms the explicitness of our version allows us to to prove two conjectures about the morphisms in the monad and we get an efficient method for machine computation of the cohomology ofsheaves we also construct all the monads for a sheaf that can be built from sums of line bundles and show that they are often characterized by numerical data let v be a finite dimensional vector space over a field k and let w v be the dual space in this paper we will study complexes and resolutions over the exterior algebra e v and their relation to modules over s sym w and sheaves on projective space p w in this paper we study the bernstein gel fand gel fand bgg correspondence 1978 usually stated as an equivalence between the derived category of bounded complexes of coherent sheaves on p w and the stable category of finitely generated graded modules over e its essential content is a functor r from complexes of graded s modules to complexes of graded e modules and its adjoint l for example if m imi is a graded s module regarded as a complex with just one term then the first and third authors are grateful to the nsf for partial support during the preparation of this paper the third author wishes to thank msri for its hospitality

von neumann entropy penalization and low rank matrix estimation we study a problem of estimation of a hermitian nonnegatively definite matrix of unit trace for instance a density matrix of a quantum system based on n i i d measurements x1 y1 xn yn where

tame and wild projective curves and classification of vector bundles abstract we propose a new method of classifying vector bundles on projective curves especially singular ones according to their representation type in particular we prove that the classification problem of vector bundles respectively of torsion free sheaves on projective curves is always finite tame or wild we completely classify curves which are of finite respectively tame vector bundle type by their dual graph moreover our methods yield a geometric description of all indecomposable vector bundles and torsion free sheaves on finite and tame curves

three themes of syzygies we present three exciting themes of syzygies where major progress was made recently boij soderberg theory stillman s question and syzygies over complete intersections

flips and the hilbert scheme over an exterior algebra we study the tangent space at a monomial point m in the hilbert scheme that parameterizes all ideals with the same hilbert function as m over an exterior algebra

bezoutians and tate resolutions abstract this paper gives an explicit construction of the tate resolution of sheaves arising from the d fold veronese embedding of p n our description involves the bezoutian of n 1 homogeneous forms of degree d in n 1 variables we give applications to duality theorems including koszul duality

schubert complexes and degeneracy loci the classical thom porteous formula expresses the homology class of the degeneracy locus of a generic map between two vector bundles as an alternating sum of schur polynomials a proof of this formula was given by pragacz by expressing this alternating sum as the euler characteristic of a schur complex which gives an explanation for the signs fulton later generalized this formula to the situation of flags of vector bundles by using alternating sums of schubert polynomials building on the schubert functors of kraand pragacz we introduce schubert complexes and show that fulton s alternating sum can be realized as the euler characteristic of this complex thereby providing a conceptual proof for why an alternating sum appears

betti numbers of determinantal ideals abstract let r k x 1 x n be a polynomial ring and let i r be a graded ideal in t romer betti numbers and shifts in minimal graded free resolutions arxiv ac 070119 romer asked whether under the cohen macaulay assumption the ith betti number i r i can be bounded above by a function of the maximal shifts in the minimal graded free r resolution of r i as well as bounded below by a function of the minimal shifts the goal of this paper is to establish such bounds for graded cohen macaulay algebras k x 1 x n i when i is a standard determinantal ideal of arbitrary codimension we also discuss other examples as well as when these bounds are sharp

statistics of equally weighted random paths on a class of self similar structures we study the statistics of equally weighted random walk paths on a family of sierpinski gasket lattices whose members are labelled by an integer b 2 b 2 mean path end to end distance grows more slowly than any power of its length n we provide arguments for the emergence of usual power law critical behaviour in the limit b when fractal lattices become almost compact