on the generalized scarf complex of lattice ideals abstract let k be a field l z n be a lattice such that l n n 0 and i l r k x 1 x n the corresponding lattice ideal we present the generalized scarf complex of i l and show that it is indispensable in the sense that it is contained in every minimal free resolution of r i l

deformation of characteristic curves of the plane ideal plasticity equations by point symmetries abstract the hyperbolic system of plane ideal plasticity equations under the saint venant mises yield criterion is considered its characteristics curves are deformed by the action of admitted group of point transformations that permits to construct a new analytical solution the mechanical sense of obtained characteristic fields is discussed the general algorithm of the relation of solutions of quasilinear hyperbolic system of two homogeneous equations of two independent variables is proposed

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

on birational macaulayfications and cohen macaulay canonical modules abstract the aim of the paper is twofold at first there is a characterization of those local domains admitting a birational macaulayfication it turns out that a local domain a quotient of a gorenstein ring possesses a cohen macaulay birational extension b if and only if its canonical module k a is a cohen macaulay module in this situation it follows that b is isomorphic to the endomorphism ring of k a the second part of the paper is devoted to the question when the canonical module of a local ring is a cohen macaulay module there are applications on sequentially cohen macaulay rings a large class of rings whose canonical module are cohen macaulay is given by the simplicial affine semigroup rings as a technical tool there is a spectral sequence related to the duality with respect to the dualizing complex

a vanishing theorem for log canonical pairs using inversion of adjunction we deduce from nadel s theorem a vanishing property for ideals sheaves on projective varieties a special case of which recovers a result due to bertram ein lazarsfeld this enables us to generalize to a large class of projective schemes certain bounds on castelnuovo mumford regularity previously obtained by bertram ein lazarsfeld in the smooth case and by chardin ulrich for locally complete intersection varieties with rational singularities our results are tested on several examples

castelnuovo mumford regularity bounds for singular surfaces we prove the regularity conjecture namely eisenbud goto conjecture for a normal surface with rational gorenstein elliptic and log canonical singularities along the way we bound the regularity for a dimension zero scheme by its loewy length and for a curve allowing embedded or isolated point components by its arithmetic degree

smooth threefolds in bbb p 5 without apparent triple or quadruple points and a quadruple point formula for a projective variety x of codimension 2 in bbb p n 2 defined over the complex number field bbb c it is traditionally said that x has no apparent k 1 ple points if the k 1 secant lines of x do not fill up the ambient projective space bbb p n 2 equivalently the locus of k 1 ple points of a generic projection of x to bbb p n 1 is empty we show that a smooth threefold in bbb p 5 has no apparent triple points if and only if it is contained in a quadric hypersurface we also obtain an enumerative formula counting the quadrisecant lines of x passing through a general point of bbb p 5 and give necessary cohomological conditions for smooth threefolds in bbb p 5 without apparent quadruple points this work is intended to generalize the work of f severi fse and a aure au where it was shown that a smooth surface in bbb p 4 has no triple points if and only if it is either a quintic elliptic scroll or contained in a hyperquadric furthermore we give open questions along these lines

symbolic powers of sums of ideals let i and j be nonzero ideals in two noetherian algebras a and b over a field k let i j denote the ideal generated by i and j in a otimes k b we prove the following expansion for the symbolic powers begin aligned i j n sum i j n i i j j end aligned if a and b are polynomial rings and if mathrm char k 0 or if i and j are monomial ideals we give exact formulas for the depth and the castelnuovo mumford regularity of i j n which depend on the interplay between the symbolic powers of i and j the proof involves a result of independent interest which states that the induced map mathrm tor i a k i n rightarrow mathrm tor i r k i n 1 is zero for any homogeneous ideal i and i ge 0 n ge 0 we also investigate other properties and invariants of i j n such as the equality between ordinary and symbolic powers the waldschmidt constant and the cohen macaulayness

castelnuovo mumford regularity of simplicial toric rings abstract bounds for the castelnuovo mumford regularity of simplicial toric rings are given which are close to the bound stated in eisenbud goto s conjecture

a vanishing theorem and asymptotic regularity of powers of ideal sheaves abstract let i be an ideal sheaf on p n in the first part of this paper we bound the asymptotic regularity of powers of i as s p reg i p s p e where e is a constant and s is the s invariant of i we also give the same upper bound for the asymptotic regularity of symbolic powers of i under some conditions in the second part by using multiplier ideal sheaves we give a vanishing theorem of powers of i when it defines a local complete intersection subvariety with log canonical singularities

b weyl spectrum and poles of the resolvent abstract let t be a bounded linear operator acting on a banach space and let bw t c such that t i is not a b fredholm operator of index 0 be the b weyl spectrum of t define also e t to be the set of all isolated eigenvalues in the spectrum t of t and t to be the set of the poles of the resolvent of t in this paper two new generalized versions of the classical weyl s theorem are considered more precisely we seek for conditions under which an operator t satisfies the generalized weyl s theorem bw t t e t or the version ii of the generalized weyl s theorem bw t t t

castelnuovo mumford regularity of projected roth varieties abstract we work over an algebraically closed field of characteristic zero the purpose of this paper is to study the castelnuovo mumford regularity of projected roth varieties which are divisors of rational scrolls of certain types these varieties appear as an exceptional case in the study of finding whether a projective variety is defined by equations of the expected low degree

a note on generalized weyl s theorem abstract we prove that if either t or t has the single valued extension property then the spectral mapping theorem holds for b weyl spectrum if moreover t is isoloid and generalized weyl s theorem holds for t then generalized weyl s theorem holds for f t for every f h t an application is given for algebraically paranormal operators

sign conditions for the existence of at least one positive solution of a sparse polynomial system abstract we give sign conditions on the support and coefficients of a sparse system of d generalized polynomials in d variables that guarantee the existence of at least one positive real root based on degree theory and gale duality in the case of integer exponents we relate our sufficient conditions to algebraic conditions that emerged in the study of toric ideals

a combinatorial proof of the eisenbud goto conjecture for monomial curves and some simplicial semigroup rings abstract we will give a pure combinatorial proof of the eisenbud goto conjecture for arbitrary monomial curves moreover we will show that the conjecture holds for certain simplicial affine semigroup rings

liaison and castelnuovo mumford regularity in this article we establish bounds for the castelnuovo mumford regularity of projective schemes in terms of the degrees of their defining equations the main new ingredient in our proofs is to show that generic residual intersections of complete intersection rational singularities again have rational singularities when applied to the theory of residual intersections this circle of ideas also sheds new light on some known classes of free resolutions of residual ideals

generic projections the equations defining projective varieties and castelnuovo regularity for a reduced irreducible projective variety x of degree d and codimension e in mathbb p n the castelnuovo mumford regularity rm reg x is defined as the least k such that x is k regular i e h i mathbb p n mathcal i x k i 0 for i ge1 where mathcal i x subset cal o mathbb p n is the sheaf of ideals of x there is a long standing conjecture about k regularity see 5 rm reg x le d e 1 here we show that rm reg x le d e 1 10 for any smooth fivefold and rm reg x le d e 1 20 for any smooth sixfold by extending methods used in 10 furthermore we give a bound for the regularity of a reduced connected and equidimensional locally cohen macaulay curve or surface in terms of degree d codimension e and an arithmetic genus rho a see theorem 4 1

toric hilbert schemes we introduce and study the toric hilbert scheme that parametrizes all ideals with the same multigraded hilbert function as a given toric ideal

initial ideals veronese subrings and rates of algebras abstract let s be a polynomial ring over an infinite field and let i be a homogeneous ideal of s let t d be a polynomial ring whose variables correspond to the monomials of degree d in s we study the initial ideals of the ideals v d i t d that define the veronese subrings of s i in suitable orders they are easily deduced from the initial ideal of i we show that in v d i is generated in degree max reg i d 2 where reg i is the regularity of the ideal i in other words the d th veronese subrings of any commutative graded ring s i has a grobner basis of degree max i d 2 we also give bounds on the regularity of i in terms of the degrees of the generators of in i and some combinatorial data this implies a version of backelin s theorem that high veronese subrings of any ring are homogeneous koszul algebras in the sense of priddy trans amer math soc 152 1970 39 60 we also give a general obstruction for a homogeneous ideal i s to have an initial ideal in i that is generated by quadrics beyond the obvious requirement that i itself should be generated by quadrics and the stronger statement that s i is koszul we use the obstruction to show that in certain dimensions a generic complete intersection of quadrics cannot have an initial ideal that is generated by quadrics for the application to backelin s theorem we require a result of backelin whose proof has never appeared we give a simple proof of a sharpened version bounding the rate of growth of the degrees of generators for syzygies of any multihomogenous module over a polynomial ring modulo an ideal generated by monomials following a method of bruns and herzog

symmetry group analysis of an ideal plastic flow in this paper we study a finite dimensional lie point symmetry group of a system describing an ideal plastic plane flow in two dimensions in order to find analytical solutions the infinitesimal generators that span this lie algebra are given we completely classify the subalgebras of codimension up to two into conjugacy classes under the action of the symmetry group based on invariant forms we use ansatze to compute symmetry reductions in such a way that the obtained solutions simultaneously cover many invariant and partially invariant solutions we calculate solutions of algebraic trigonometric inverse trigonometric and elliptic type some solutions depending on one or two arbitrary functions of one variable have also been found in some cases the shape of a potentially feasible extrusion die corresponding to the solution is deduced these tools could be used to thin curve undulate or shape a ring in an ideal plastic material

the regularity of partial elimination ideals castelnuovo normality and syzygies abstract let x be a reduced closed subscheme in p n x x p n 1 be a projection from a point outside x and z i x x be the closed subscheme defined by the i th partial elimination ideal k i i x which is supported on the i 1 th multiple points of in this paper motivated from projection methods to prove eisenbud goto conjecture on regularity in many cases we describe the syzygetic behaviors and castelnuovo normality of the projection with a viewpoint of the regularity of the partial elimination ideal k i i x i 1 or that of the multiple locus z i x of we also give some applications to the syzygies and castelnuovo normality of successive projections which recover and generalize some known results in 1 3 15 16

the regularity of a toric variety abstract we give a method for computing the degrees of the minimal syzygies of a toric variety by means of combinatorial techniques indeed we complete the explicit description of the minimal free resolution of the associated semigroup algebra using the simplicial representation of koszul homology which appeared in a campillo and c marijuan 1991 sem theor nombres bordeaux3 249 260 as an application we obtain an algorithm for computing the castelnuovo mumford regularity of a projective toric variety this regularity is explicitly bounded by means of the semigroup generators which parametrize the variety

generic initial ideals of singular curves in graded lexicographic order abstract in this paper we are interested in the generic initial ideals of singular projective curves with respect to the graded lexicographic order let c be a singular irreducible projective curve of degree d 5 with the arithmetic genus a c in p r where r 3 if m i c is the regularity of the lexicographic generic initial ideal of i c in a polynomial ring k x 0 x r then we prove that m i c is 1 d 1 2 a c which is obtained from the monomial x r 3 x r 1 d 1 2 a c provided that dim tan p c 2 for every singular point p c this number is equal to one plus the number of secant lines through the center of general projection into p 2 our result generalizes the work of j ahn 2008 1 for smooth projective curves and that of a conca and j sidman 2005 9 for smooth complete intersection curves in p 3 the case of singular curves was motivated by a conca and j sidman 2005 9 example 4 3 we also provide some illuminating examples of our results via calculations done with macaulay 2 and singular decker et al 2011 10 grayson and stillman 16

graded mapping cone theorem multisecants and syzygies abstract let x be a reduced closed subscheme in p n as a slight generalization of property n p due to green lazarsfeld one says that x satisfies property n 2 p scheme theoretically if there is an ideal i generating the ideal sheaf i x p n such that i is generated by quadrics and there are only linear syzygies up to p th step cf eisenbud et al 2005 8 vermeire 2001 20 recently many algebraic and geometric results have been proved for projective varieties satisfying property n 2 p cf choi kwak and park 2008 6 eisenbud et al 2005 8 kwak and park 2005 15 in this case the castelnuovo regularity and normality can be obtained by the blowing up method as reg x e 1 where e is the codimension of a smooth variety x cf bertram ein and lazarsfeld 2003 3 on the other hand projection methods have been very useful and powerful in bounding castelnuovo regularity normality and other classical invariants in geometry cf beheshti and eisenbud 2010 2 kwak 1998 14 kwak and park 2005 15 lazarsfeld 1987 16 we first prove the graded mapping cone theorem on partial eliminations as a general algebraic tool to study syzygies of the non complete embedding of x for applications we give an optimal bound on the length of zero dimensional intersections of x and a linear space l in terms of graded betti numbers we also deduce several theorems about the relationship between x and its projections with respect to the geometry and syzygies for a projective scheme x satisfying property n 2 p scheme theoretically in addition we give not only interesting information on the regularity of fibers of the projection for the case of n d p d 2 but also geometric structures for projections according to moving the center

some symmetry group aspects of a perfect plane plasticity system in this paper all the known classical solutions of a plane perfect plasticity system under the saint venant tresca von mises yield criterion are associated with some group of point symmetries the equations of slip line families for all solutions are constructed which allows one to explicitly determine the boundaries of the plastic areas it is shown how one can determine the compatible velocity solution for known stresses by considering symmetries some invariant solutions of velocities for prandtl stresses are constructed the mechanical sense of the obtained velocity fields is discussed to the blessed memory of our teacher d d ivlev

a castelnuovo mumford regularity bound for scrolls abstract let x p r be a scroll of codimension e and degree d over a smooth projective curve of genus g the purpose of this paper is to prove a linear castelnuovo mumford regularity bound that reg x d e 1 g e 1 this bound works over an algebraically closed field of arbitrary characteristic

on a theorem of castelnuovo and the equations defining space curves introduction 491 w notat ion and conventions 493 w a regularity theorem 494 w a rationality theorem 498 w the existence of secant lines 501 w open problems 505 references 506

on the complexity of the integral closure the computation of the integral closure of an affine ring has been the focus of several modern algorithms we will treat here one related problem the number of generators the integral closure of an affine ring may require this number and the degrees of the generators in the graded case are major measures of cost of the computation we prove several polynomial type bounds for various kinds of algebras and establish in characteristic zero an exponential type bound for homogeneous algebras with a small singular locus

symmetry groups of non stationary planar ideal plasticity this paper is a study of the lie groups of point symmetries admitted by a system describing a non stationary planar flow of an ideal plastic material for several types of forces involved in the system the infinitesimal generators which generate the lie algebra of symmetries have been obtained in the case of a monogenic force the classification of one and two dimensional subalgebras into conjugacy classes under the action of the group of automorphisms has been accomplished the method of symmetry reduction is applied for certain subalgebra classes in order to obtain invariant solutions

vanishing theorems a theorem of severi and the equations defining projective varieties in 1903 severi s proved that if x x c p are smooth surfaces whose union is the complete intersection of r 2 hypersurfaces of degrees di dr 2 then hypersurfaces of degree k z di r cut out a complete linear series on x cf sr xiii 9 8 the purpose of this paper is first of all to show that elementary arguments using the kodaira vanishing theorem lead to a simple variant of severi s statement theorem 7 below which extends it in serval directions more importantly we hope to convince the reader that this result has a surprising number of applications to questions involving the equations defining projective varieties consider to begin with a smooth complex projective variety x c pr of dimension n and codimension e r n in this setting our theorem asserts the vanishing of the higher cohomology of suitable twists of the ath power of the ideal sheaf yx of x in apr proposition 1 assume that x is cut out scheme theoretically in p by hypersurfaces of degrees d1 d2 d then h p r k 0 for i 1 provided k ad d2 de r note that only the degrees of the first e codim x pr defining equations come into play here when n 2 and a 1 this is a consequence of severi s

on the genus and hartshorne rao module of projective curves in this paper optimal upper bounds for the genus and the dimension of the graded components of the hartshorne rao module of curves in projective n space are established this generalizes earlier work by hartshorne h and martin deschamps and perrin mdp special emphasis is put on curves in bf p 4 the first main result is a so called restriction theorem it says that a non degenerate curve of degree d geq 4 in bf p 4 over a field of characteristic zero has a non degenerate general hyperplane section if and only if it does not contain a planar curve of degree d 1 see th 1 3 then using methods of brodmann and nagel bounds for the genus and hartshorne rao module of curves in bf p n with non degenerate general hyperplane section are derived it is shown that these bounds are best possible in a very strict sense coupling these bounds with the restriction theorem gives the second main result for curves in bf p 4 then curves of maximal genus are investigated the betti numbers of their minimal free resolutions are computed and a description of all reduced curves of maximal genus in bf p n of degree geq n 2 is given finally all pairs d g of integers which really occur as the degree d and genus g of a non degenerate curve in bf p 4 are described

property gb and perturbations abstract an operator t acting on a banach space x possesses property gb if a t sbf t t where a t is the approximate point spectrum of t sbf t is the essential semi b fredholm spectrum of t and t is the set of all poles of the resolvent of t in this paper we study property gb in connection with weyl type theorems which is analogous to generalized browder s theorem several sufficient and necessary conditions for which property gb holds are given we also study the stability of property gb for a polaroid operator t acting on a banach space under perturbations by finite rank operators by nilpotent operators and more generally by algebraic and riesz operators commuting with t

on the depth of the associated graded ring of an ideal abstract let r be a local cohen macaulay ring let i be an r ideal and let g be the associated graded ring of i we give an estimate for the depth of g when g fails to be cohen macaulay we assume that i has a small reduction number and sufficiently good residual intersection properties and satisfies local conditions on the depth of some powers the main theorem unifies and generalizes several known results we also give conditions that imply the serre properties of the blow up rings

bouquet algebra of toric ideals abstract to any toric ideal i a encoded by an integer matrix a we associate a matroid structure called the bouquet graph of a and introduce another toric ideal called the bouquet ideal of a we show how these objects capture the essential combinatorial and algebraic information about i a passing from the toric ideal to its bouquet ideal via the graph theoretic properties of the bouquet graph allows us to classify several cases for example on the one end of the spectrum there are ideals that we call stable for which bouquets capture the complexity of various generating sets as well as the minimal free resolution on the other end of the spectrum lie toric ideals whose various bases e g minimal generating sets grobner graver bases coincide apart from allowing for classification type results bouquets provide a new way to construct families of examples of toric ideals with various interesting properties such as robustness genericity and unimodularity the new bouquet framework can be used to provide a characterization of toric ideals whose graver basis the universal grobner basis any reduced grobner basis and any minimal generating set coincide

regularity of prime ideals we answer several natural questions which arise from a recent paper of mccullough and peeva providing counterexamples to the eisenbud goto regularity conjecture we give counterexamples using rees algebras and also construct counterexamples that do not rely on the mayr meyer construction furthermore examples of prime ideals for which the difference between the maximal degree of a minimal generator and the maximal degree of a minimal first syzygy can be made arbitrarily large are given using a result of ananyan hochster we show that there exists an upper bound on regularity of prime ideals in terms of the multiplicity alone

a sharp bound for the castelnuovo mumford regularity of subspace arrangements abstract we show that the ideal of an arrangement of d linear subspaces of projective space is d regular in the sense of castelnuovo and mumford answering a question of b sturmfels in particular this implies that the ideal of an arrangement of d subspaces is generated in degrees less than or equal to d

property w and perturbations ii abstract a bounded linear operator t l x defined on a banach space x satisfies property w a variant of weyl s theorem if the complement in the approximate point spectrum a t of the weyl essential approximate spectrum wa t coincides with the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity in this note we study the stability of property w for a bounded operator t acting on a banach space under perturbations by finite rank operators by nilpotent operator and quasi nilpotent operators commuting with t

singularities of generic linkage of algebraic varieties let y be a generic link of a subvariety x of a nonsingular variety a we give a description of the grauert riemenschneider canonical sheaf of y in terms of the multiplier ideal sheaves associ ated to x and use it to study the singularities of y as the first application we give a criterion when y has rational singularities and show that log canonical threshold increases and log canonical pairs are preserved in generic linkage as another application we give a quick and simple liaison method to generalize the results of de fernex ein and chardin ulrich on the castelnuovo mumford regularity bound for a projective variety

on the castelnuovo mumford regularity of connected curves in this paper we prove that the regularity of a connected curve is bounded by its degree minus its codimension plus 1 we also investigate the structure of connected curves for which this bound is optimal in particular we construct connected curves of arbitrarily high degree in p 4 having maximal regularity but no extremal secants we also show that any connected curve in p 3 of degree at least 5 with maximal regularity and no linear components has an extremal secant

syzygies of affine toric varieties we give a method for computing the syzygies of the coordinate ring r of an affine toric variety we show how the method works for dimension one and two cases cohen macaulay semigroups and for computing minimal generators of the defining ideal we show how to compute the depth of r and generalize a criterion for cohen macaulayness

castelnuovo mumford regularity of seminormal simplicial affine semigroup rings abstract we show that the eisenbud goto conjecture holds for homogeneous seminormal simplicial affine semigroup rings moreover we prove an upper bound for the castelnuovo mumford regularity in terms of the dimension which is similar as in the normal case finally we compute explicitly the regularity of full veronese rings

inversion of adjunction for local complete intersection varieties we extend inversion of adjunction and semicontinity of minimal log discrepancies from the case of a smooth ambient variety to the case of local complete intersection varieties

variations on weyl s theorem in this note we study the property w a variant of weyl s theorem introduced by rakocevic by means of the localized single valued extension property svep we establish for a bounded linear operator defined on a banach space several sufficient and necessary conditions for which property w holds we also relate this property with weyl s theorem and with another variant of it a weyl s theorem we show that weyl s theorem a weyl s theorem and property w for t respectively t coincide whenever t respectively t satisfies svep as a consequence of these results we obtain that several classes of commonly considered operators have property w

rational normal scrolls and the defining equations of rees algebras consider a height two ideal i which is minimally generated by m homogeneous forms of degree d in the polynomial ring r k x y suppose that one column in the homogeneous presenting matrix of i has entries of degree n and all of the other entries of are linear we identify an explicit generating set for the ideal a which defines the rees ring r r it so r s a for the polynomial ring s r t1 tm we resolve r as an s module and is as an r module for all powers s the proof uses a rational normal scroll ring a s h with aa isomorphic to the nth symbolic power of a height one prime ideal k of a the ideal k n is generated by monomials whenever possible we study a k n in place of a aa because the generators of k n are much less complicated then the generators of aa we obtain a filtration of k n in which the factors are polynomial rings hypersurface rings or modules resolved by eagon northcott complexes the generators of i give rise to an algebraic curve c in projective m 1 space the defining equations of the fiber ring r x y r yield a solution of the implicitization problem for c introduction let i be a height two ideal in the polynomial ring r k x y with i minimally generated by m forms of degree d we consider the following questions what is the defining ideal a of the rees ring r r it r it it what is the hilbert function of r the rees ring r is equal to r t1 tm a what is the minimal resolution of r what are the defining equations of the fiber 1supported in part by the national security agency 2supported in part by the national science foundation and the national security agency 3supported in part by the national science foundation 2000 mathematics subject classification primary 14q05 secondary 13c20 13d02 14e05 14m12 68w30

base point freeness of double point divisors of smooth birational divisors on conical rational scrolls abstract we work over an algebraically closed field of characteristic zero the purpose of this paper is to prove that the complete linear system of the double point divisors of smooth birational divisors on conical rational scrolls are base point free a smooth birational divisor on a conical rational scroll has a nonbirational inner center that is a point on it from which the linear projection gives nonbirational map to its image in the previous paper by the author it was shown that for a projective variety without nonbirational inner centers its double point divisor is base point free

ideals generated by quadrics exhibiting double exponential degrees abstract we modify the example of mayr and meyer to give an example of a polynomial ideal generated by quadrics which exhibits double exponential growth for the ideal membership problem we also show that the homogenization of this example gives a set of elements which has a minimal syzygy of double exponential degree

group analysis of an ideal plasticity model a group analysis of a system describing an ideal plastic flow is made in order to obtain analytical solutions the complete lie algebra of point symmetries of this system are given two of the infinitesimal generators that span the lie algebra are original to this paper a classification into conjugacy classes of all one and two dimensional subalgebras is performed invariant and partially invariant solutions corresponding to certain conjugacy classes are obtained using the symmetry reduction method solutions of algebraic trigonometric inverse trigonometric and elliptic type are provided as illustrations and other solutions expressed in terms of one or two arbitrary functions have also been found for some of these solutions a physical interpretation allows one to determine the shape of feasible extrusion dies corresponding to these solutions the corresponding tools could be used to curve rods or slabs or to shape a ring in an ideal plastic material by an extrusion process

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

property w for perturbations of polaroid operators abstract a bounded linear operator t l x acting on a banach space satisfies property w a variant of weyl s theorem if the complement in the approximate point spectrum a t of the weyl essential approximate point spectrum wa t is the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity in this note we study the stability of property w for a polaroid operator t acting on a banach space under perturbations by finite rank operators by nilpotent operators and more generally by algebraic operators commuting with t

a bound for castelnuovo mumford regularity by double point divisors abstract let x p r be a non degenerate smooth projective variety of dimension n codimension e and degree d defined over an algebraically closed field of characteristic zero in this paper we first show that reg o x d e and classify the extremal and the next to extremal cases our result reduces the eisenbud goto regularity conjecture for the smooth case to the problem finding a castelnuovo type bound for normality it is worth noting that mccullough peeva recently constructed counterexamples to the regularity conjecture by showing that reg o x is not even bounded above by any polynomial function of d when x is not smooth for a normality bound in the smooth case we establish that reg x n d 2 1 which improves previous results obtained by mumford bertram ein lazarsfeld and noma finally by generalizing mumford s method on double point divisors we prove that reg x d 1 m where m is an invariant arising from double point divisors associated to outer general projections using double point divisors associated to inner projections we also obtain a slightly better bound for reg x under suitable assumptions

regularity of smooth curves in biprojective spaces abstract maclagan and smith d maclagan g g smith multigraded castelnuovo mumford regularity j reine angew math 571 2004 179 212 developed a multigraded version of castelnuovo mumford regularity based on their definition we will prove in this paper that for a smooth curve c p a p b a b 2 of bidegree d 1 d 2 with nondegenerate birational projections the ideal sheaf i c p a p b is d 2 b 1 d 1 a 1 regular we also give an example showing that in some cases this bound is the best possible

the complexity of the word problems for commutative semigroups and polynomial ideals abstract any decision procedure for the word problems for commutative semigroups and polynomial deals inherently requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied this bound is achieved by a simple procedure for the semigroup problem