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

n point deformation of algebraic k3 surfaces we construct a set of noncommutative geometries by performing n point deformation of algebraic k3 surfaces first we consider two point deformation of algebraic k3 surfaces by performing algebraic deformation of a pair of commutative algebraic k3 surfaces in this case the moduli space of the noncommutative deformations is of dimension 19 the same as the moduli dimension of the complex deformations of commutative algebraic k3 surfaces then we extend this method to the n point case in the n point case the dimension of deformation moduli space becomes 19n n 1 2

the classification of homogeneous cohen macaulay rings of finite representation type in this paper we classify the homogeneous cohen macaulay rings which are of finite representation type that is the cohen macaulay rings which are positively graded and generated in degree 1 with an algebraically closed field of characteristic 0 in degree 0 and which have up to isomorphisms and shifts in the grading only a finite number of indecomposable maximal cohen macaulay modules mcm modules our main contribution is to show that a homogeneous cohen macaulay ring of finite representation type and dimension 2 must have minimal multiplicity putting this together with previous results of auslander auslander reiten buchweitz greuel schreyer greuel kn6rrer and solberg we obtain

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

generalizations of the fkkm theorem and the ky fan minimax inequality with applications to maximal elements price equilibrium and complementarity abstract this paper generalizes the fan knaster kuratowski mazurkiewicz fkkm theorem of ky fan game theory and related topics pp 151 156 north holland amsterdam 1979 and math ann 266 1984 519 537 and the ky fan minimax inequality by introducing a class of the generalized closedness and continuity conditions which are called the transfer closedness and transfer continuities we then apply these results to prove the existence of maximal elements of binary relations under very weak assumptions we also prove the existence of price equilibrium and the complementarity problem without the continuity assumptions thus our results generalize many of the existence theorems in the literature

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

rank 3 arithmetically cohen macaulay bundles over hypersurfaces abstract let x be a smooth projective hypersurface of dimension 5 and let e be an arithmetically cohen macaulay bundle on x of any rank we prove that e splits as a direct sum of line bundles if and only if h i x 2 e 0 for i 1 2 3 4 as a corollary this result proves a conjecture of buchweitz greuel and schreyer for the case of rank 3 arithmetically cohen macaulay bundles

some problems and results in the study of nonlinear analysis this paper is presented as a survey of the latest and new results on some topics in the study of nonlinear analysis which were obtained by the author and some chinese mathematicians

surface singularities of finite buchsbaum representation type the purpose of this paper is to determine the structure of surface singularities of finite buchsbaum representation type and the main results are summarized into the following let r be a noetherian complete local ring of dim r 2 and assume that the residue class field of r is algebraically closed let e r and v r denote respectively the multiplicity and the embedding dimension of r then the following three conditions are equivalent 1 r has a finite buchsbaum representation type that is r possesses only finitely many isomorphism classes of indecomposable maximal buchsbaum r modules 2 e r 1 and v r 3 3 r p xi where p is a three dimensional complete regular local ring with maximal ideal n x n n 2 and i is an ideal of p such that ht p i 2

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

rank 2 arithmetically cohen macaulay bundles on a nonsingular cubic surface abstract rank 2 indecomposable arithmetically cohen macaulay bundles e on a nonsingular cubic surface x in p 3 are classified by means of the possible forms taken by the minimal graded free resolution of e over p 3 the admissible values of the chern classes of e are listed and the vanishing locus of a general section of e is studied properties of e such as slope semi stability and simplicity are investigated the number of relevant families is computed together with their dimension

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

indecomposable generalized cohen macaulay modules the aim of this paper is to study the indecomposable modules which are cohen macaulay on the punctured spectrum of a local ring and to describe some of their invariants such as their local cohomology groups and ranks one of our main concerns is to find indecomposable quasi buchsbaum modules of high rank with prescribed cohomology over a regular local ring

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

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

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

regularity and cohomological splitting conditions for vector bundles on multiprojective spaces abstract we give a definition of regularity on multiprojective spaces which is different from the definitions of hoffman and wang and costa and miro roig using this notion we prove some splitting criteria for vector bundles

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

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

spinor bundles on quadrics we define some stable vector bundles on the complex quadric hypersurface qn of dimension n as the natural generalization of the universal bundle and the dual of the quotient bundle on q4 gr l 3 we call them spinor bundles when n 2fc 1 there is one spinor bundle of rank 2k 1 when n 2k there are two spinor bundles of rank 2k 1 their behavior is slightly different according as n 0 mod 4 or n 2 mod 4 as an application we describe some moduli spaces of rank 3 vector bundles on q5 and qe introduction let qn be the smooth quadric hypersurface of the complex pro jective space pn 1 in this paper we define in a geometrical way some vector bundles on the quadric qn as the natural generalization of the universal bundle and the dual of the quotient bundle on q4 gr l 3 we call them spinor bundles on q4 this definition is equivalent to the usual one spinor bundles are homogeneous and stable according to the definition of mum ford takemoto we study their first properties using the geometrical description given and some standard techniques available in oss we also use a theorem of ramanan see um about the stability of homoge neous bundles induced by irreducible representations when n is odd there is only one spinor bundle while when n is even there are two nonisomorphic spinor bun dles when n is even the behavior of spinor bundles is slightly different according as n 0 mod4 or n 2 mod4 in ot2 we have given a cohomological splitting criterion for vector bundles on quadrics involving spinor bundles qn spin n 2 p cty st is a homogeneous manifold and the semisimple part of the lie algebra of p ai is o n at the level of lie algebras spinor bundles are defined from the spin and half spin representations of o n the paper is divided into three sections in 1 we give some preliminary results and we define the spinor bundles in 2 we study the first properties of spinor bundles in 3 as an application we describe some moduli spaces of rank 3 vector bundles on q5 and qq

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

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

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

the minimal resolution conjecture abstract by extending the ideal generation conjecture we formulate the minimal resolution conjecture for a sufficiently generic set of points in p n and we explicitly write out the conjectured values of the betti numbers we then relate the minimal resolution conjecture to the ideal generation conjecture and the cohen macaulay type conjecture and we prove the minimal resolution conjecture for d nn 1 n points for n 2 points and for d 1 nn points we also prove the cohen macaulay type conjecture for s points with n 1 s 2 nn finally we recover information about the resolution of s 1 or s 1 points from that of s points

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

on ky fan s section theorem abstract in this paper we first give a new section theorem of ky fan type and its equivalent theorem in an h space and next as their applications we obtain some minimax theorems and a coincidence theorem

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

supersymmetric quantum theory and non commutative geometry classical differential geometry can be encoded in spectral data such as connes spectral triples involving supersymmetry algebras in this paper we formulate non commutative geometry in terms of supersymmetric spectral data this leads to generalizations of connes non commutative spin geometry encompassing non commutative riemannian symplectic complex hermitian and hyper kahler geometry a general framework for non commutative geometry is developed from the point of view of supersymmetry and illustrated in terms of examples in particular the non commutative torus and the non commutative 3 sphere are studied in some detail

test modules to calculate dutta multiplicities using frobenius maps the dutta multiplicity f was first defined by dutta for a complex f over a local ring of positive characteristic nowadays using localized chern characters or adams operations it is defined for a complex not only over a local ring of positive characteristic but also over a ring that is a homomorphic image of an arbitrary regular local ring in the paper we shall give another characterization to dutta multiplicity using some galois extension furthermore we shall define the notion of test modules if a test module for a local ring exists then the positivity conjecture of dutta multiplicities is true over the ring in the paper it is proved that if the small cohen macaulay modules conjecture is true then test modules always exist for complete local domains whose field of fractions are of characteristic 0

matrix theory compactification on noncommutative t4 z2 in this article we construct gauge bundles on a noncommutative toroidal orbifold t 4 z2 first we explicitly construct a bundle with constant curvature connections on a noncommutative t 4 following rieffel s method then applying the appropriate quotient conditions for its z2 orbifold we find a connes douglas schwarz type solution of matrix theory compactified on t 4 z2 when we consider two copies of a bundle on t 4 invariant under the z2 action the resulting higgs branch moduli space of equivariant constant curvature connections becomes an ordinary toroidal orbifold t4 z2

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

theta functions on noncommutative t4 we construct the so called theta vectors on noncommutative t4 which correspond to the theta functions on commutative tori with complex structures following the method of dieng and schwarz we first construct holomorphic connections and then find the functions satisfying the holomorphic conditions the theta vectors the holomorphic structure in the noncommutative t4 case is given by a 2 2 complex matrix and the consistency requires its off diagonal elements to be the same we also construct the tensor product of these functions satisfying the consistency requirement

super stretched and graded countable cohen macaulay type abstract we define what it means for a cohen macaulay ring to be super stretched and show that cohen macaulay rings of graded countable cohen macaulay type are super stretched we use this result to show that rings of graded countable cohen macaulay type and positive dimension have possible h vectors 1 1 n or 1 n 1 further one dimensional standard graded gorenstein rings of graded countable type are shown to be hypersurfaces this result is not known in higher dimensions in the non gorenstein case rings of graded countable cohen macaulay type of dimension larger than 2 are shown to be of minimal multiplicity

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

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

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

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

on a form of kkm principle and supinfsup inequalities of von neumann and of ky fan type abstract a version of kkm principle is considered together with its application to supinfsup inequalities of von neumann and of ky fan type these inequalities which are respectively generalizations of von neumann minimax theorem and minimax inequality of ky fan provide new light in obtaining either new results or new proofs for known results

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

an intersection theorem in l convex spaces with applications a new intersection theorem is obtained in l convex spaces without linear structure as its applications a fixed point theorem a maximal element theorem a coincidence theorem some new minimax inequalities and a saddle point theorem are given in l convex spaces our results generalize many known theorems in the literature

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

on the clifford algebra of a binary form the clifford algebra cj of a binary form f of degree d is the k algebra k x y i where i is the ideal generated by ax y d f e k c f has a natural homomorphic image a f that is a rank d 2 azumaya algebra over its center we prove that the center is isomorphic to the coordinate ring of the complement of an explicit divisor in pic d g 1 c where c is the curve w d f u v and g is the genus of c