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142
Gröbner geometry of Schubert polynomials
 Ann. Math
"... Schubert polynomials, which a priori represent cohomology classes of Schubert varieties in the flag manifold, also represent torusequivariant cohomology classes of certain determinantal loci in the vector space of n ×n complex matrices. Our central result is that the minors defining these “matrix S ..."
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Cited by 66 (13 self)
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Schubert polynomials, which a priori represent cohomology classes of Schubert varieties in the flag manifold, also represent torusequivariant cohomology classes of certain determinantal loci in the vector space of n ×n complex matrices. Our central result is that the minors defining these “matrix Schubert varieties” are Gröbner bases for any antidiagonal term order. The Schubert polynomials are therefore positive sums of monomials, each monomial representing the torusequivariant cohomology class of a component (a schemetheoretically reduced coordinate subspace) in the limit of the resulting Gröbner degeneration. Interpreting the Hilbert series of the flat limit in equivariant Ktheory, another corollary of the proof is that Grothendieck polynomials represent the classes of Schubert varieties in Ktheory of the flag manifold. An inductive procedure for listing the limit coordinate subspaces is provided by the proof of the Gröbner basis property, bypassing what has come to be known as Kohnert’s conjecture [Mac91]. The coordinate subspaces, which are
The Orbifold Chow Ring of Toric DeligneMumford Stacks
 JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY
, 2004
"... ..."
A generalization of tight closure and multiplier ideals
 TRANS. AM. MATH. SOC
, 2003
"... We introduce a new variant of tight closure associated to any fixed ideal a, which we call atight closure, and study various properties thereof. In our theory, the annihilator ideal τ(a) of all atight closure relations, which is a generalization of the test ideal in the usual tight closure theor ..."
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Cited by 35 (2 self)
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We introduce a new variant of tight closure associated to any fixed ideal a, which we call atight closure, and study various properties thereof. In our theory, the annihilator ideal τ(a) of all atight closure relations, which is a generalization of the test ideal in the usual tight closure theory, plays a particularly important role. We prove the correspondence of the ideal τ(a) and the multiplier ideal associated to a (or, the adjoint of a in Lipman’s sense) in normal QGorenstein rings reduced from characteristic zero to characteristic p ≫ 0. Also, in fixed prime characteristic, we establish some properties of τ(a) similar to those of multiplier ideals (e.g., a BriançonSkoda type theorem, subadditivity, etc.) with considerably simple proofs, and study the relationship between the ideal τ(a) and the Frationality of Rees algebras.
The Alexander duality functors and local duality with monomial support
 J. Algebra
"... Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated Nngraded module. The functors associated with Alexander duality provide a duality on the level of free and injective resolutions, and numerous Bass and Betti number relations result as co ..."
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Cited by 32 (13 self)
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Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated Nngraded module. The functors associated with Alexander duality provide a duality on the level of free and injective resolutions, and numerous Bass and Betti number relations result as corollaries. A minimal injective resolution of a module M is equivalent to the injective resolution of its Alexander dual, and contains all of the maps in the minimal free resolution of M over every Zngraded localization. Results are obtained on the interaction of duality for resolutions with cellular resolutions and lcmlattices. Using injective resolutions, theorems of EagonReiner and Terai are generalized to all Nngraded modules: the projective dimension of M equals the supportregularity of its Alexander dual, and M is CohenMacaulay if and only if its Alexander dual has a supportlinear free resolution. Alexander duality is applied in the context of the Zngraded local cohomology functors Hi I (−) for squarefree monomial ideals I in the polynomial ring S, proving a duality directly generalizing local duality, which is the case when I = m is maximal. In the process, a new flat complex for calculating local cohomology at monomial ideals is introduced, showing, as a consequence, that Terai’s formula for the Hilbert series of H i I (S) is equivalent to Hochster’s for Hn−i m (S/I). 1
Characteristicfree bounds for the Castelnuovo Mumford regularity
 Compos. Math
"... Abstract. In this paper we show how, given a complex of graded modules and knowing some partial CastelnuovoMumford regularities for all the modules in the complex and for all the positive homologies, it is possible to get a bound on the regularity of the zero homology. We use this to prove that if ..."
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Cited by 28 (0 self)
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Abstract. In this paper we show how, given a complex of graded modules and knowing some partial CastelnuovoMumford regularities for all the modules in the complex and for all the positive homologies, it is possible to get a bound on the regularity of the zero homology. We use this to prove that if dim TorR 1 (M, N) ≤ 1 then reg(M ⊗N) ≤ reg(M)+reg(N), generalizing results of Chandler, Conca and Herzog, and Sidman. Finally we give a description of the regularity of a module in terms of the postulation numbers of filter regular hyperplane restrictions. 1. introduction Let R = K[X1,..., Xn] be a polynomial ring over a field K, M a finitely generated graded Rmodule and let I ⊂ R be an ideal. Recently some work has been done to study when the CastelnuovoMumford regularity of Ir can be bounded by r times the regularity of I, and more generally when the regularity of IM can be bounded by the sum of the regularity of I and M. This is not always the case; see the papers of Sturmfels [St], and Conca, Herzog [CH] for counterexamples. On the
Multigraded CastelnuovoMumford Regularity
 J. REINE ANGEW. MATH
, 2003
"... We develop a multigraded variant of CastelnuovoMumford regularity. Motivated by toric ..."
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Cited by 26 (6 self)
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We develop a multigraded variant of CastelnuovoMumford regularity. Motivated by toric
The Projective Geometry of the Gale Transform
, 1998
"... The Gale transform, an involution on sets of points in projective space, appears in a multitude of guises, in subjects as diverse as optimization, coding theory, thetafunctions, and recently in our proof that certain general sets of points fail to satisfy the minimal free resolution conjecture (see ..."
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Cited by 24 (5 self)
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The Gale transform, an involution on sets of points in projective space, appears in a multitude of guises, in subjects as diverse as optimization, coding theory, thetafunctions, and recently in our proof that certain general sets of points fail to satisfy the minimal free resolution conjecture (see EisenbudPopescu [1996]). In this paper we reexamine the Gale transform in the light of modern algebraic geometry. We give a more general definition, in the context of finite (locally) Gorenstein subschemes. We put in modern form a number of the more remarkable examples discovered in the past, and we add new constructions and connections to other areas of algebraic geometry. We generalize Goppa’s theorem in coding theory and we give new applications to Castelnuovo theory. We give
Monomial and toric ideals associated to Ferrers graphs
"... Abstract. Each partition λ =(λ1,λ2,...,λn) determines a socalled Ferrers tableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed a Ferrers ideal, is a squarefree monomial ideal that is generated by quadrics. We show that such an ideal has a 2linear minimal free resolution; i.e. ..."
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Cited by 20 (2 self)
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Abstract. Each partition λ =(λ1,λ2,...,λn) determines a socalled Ferrers tableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed a Ferrers ideal, is a squarefree monomial ideal that is generated by quadrics. We show that such an ideal has a 2linear minimal free resolution; i.e. it defines a small subscheme. In fact, we prove that this property characterizes Ferrers graphs among bipartite graphs. Furthermore, using a method of Bayer and Sturmfels, we provide an explicit description of the maps in its minimal free resolution. This is obtained by associating a suitable polyhedral cell complex to the ideal/graph. Along the way, we also determine the irredundant primary decomposition of any Ferrers ideal. We conclude our analysis by studying several features of toric rings of Ferrers graphs. In particular we recover/establish formulæ for the Hilbert series, the CastelnuovoMumford regularity, and the multiplicity of these rings. While most of the previous works in this highly investigated area of research involve path counting arguments, we offer here a new and selfcontained approach based on results from Gorenstein liaison theory. 1.
Homological Methods for Hypergeometric Families
 Journal of American Mathematical Society
, 2004
"... UW dedicates this paper to the memory of his father, Hansjoachim Walther. ABSTRACT. We analyze the behavior of the holonomic rank in families of holonomic systems over complex algebraic varieties by providing homological criteria for rankjumps in this general setting. Then we investigate rankjump ..."
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Cited by 18 (11 self)
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UW dedicates this paper to the memory of his father, Hansjoachim Walther. ABSTRACT. We analyze the behavior of the holonomic rank in families of holonomic systems over complex algebraic varieties by providing homological criteria for rankjumps in this general setting. Then we investigate rankjump behavior for hypergeometric systems HA(β) arising from a d × n integer matrix A and a parameter β ∈ Cd. To do so we introduce an Euler–Koszul functor for hypergeometric families over Cd, whose homology generalizes the notion of a hypergeometric system, and we prove a homology isomorphism with our general homological construction above. We show that a parameter β ∈ Cd is rankjumping for HA(β) if and only if β lies in the Zariski closure of the set of Zdgraded degrees α where the local cohomology L i<d Hi m(C[NA])α of the semigroup ring C[NA] supported at its maximal graded ideal m is nonzero. Consequently, HA(β) has no rankjumps over Cd if and only if C[NA] is Cohen–Macaulay of dimension d. CONTENTS
Uniform bounds on multigraded regularity
 MR 2005g:14098 Zbl 1070.14006
, 2003
"... We give an effective uniform bound on the multigraded regularity of a subscheme of a smooth projective toric variety X with a given multigraded Hilbert polynomial. To establish this bound, we introduce a new combinatorial tool, called a Stanley filtration, for studying monomial ideals in the homogen ..."
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Cited by 16 (2 self)
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We give an effective uniform bound on the multigraded regularity of a subscheme of a smooth projective toric variety X with a given multigraded Hilbert polynomial. To establish this bound, we introduce a new combinatorial tool, called a Stanley filtration, for studying monomial ideals in the homogeneous coordinate ring of X. As a special case, we obtain a new proof of Gotzmann’s regularity theorem. We also discuss applications of this bound to the construction of multigraded Hilbert schemes. 1.