arXiv subjects

Claudiu Raicu

Publications and source records attributed to Claudiu Raicu.

50 records · Page 3Linked to original sources

arXiv2016

The syzygies of some thickenings of determinantal varieties

The vector space of m x n complex matrices (m >= n) admits a natural action of the group GL = GL_m x GL_n via row and column operations. For positive integers a,b, we consider the ideal I_{a x b} defined as the smallest GL-equivariant ideal containing the b-th powers of the a x a minors of the generic m x n matrix. We compute the syzygies of the ideals I_{a x b} for all a,b, together with their GL-equivariant structure, generalizing earlier results of Lascoux for the ideals of minors (b=1), and of Akin-Buchsbaum-Weyman for the powers of the ideals of maximal minors (a=n). Our methods rely on a nice connection between commutative algebra and the representation theory of the superalgebra gl(m|n), as well as on our previous calculation of Ext modules done in the context of describing local cohomology with determinantal support. Our results constitute an important ingredient in the proof by Nagpal-Sam-Snowden of the first non-trivial Noetherianity results for twisted commutative algebras which are not generated in degree one.

math.AC↗

arXiv2016

Characters of equivariant D-modules on spaces of matrices

We compute the characters of the simple GL-equivariant holonomic D-modules on the vector spaces of general, symmetric and skew-symmetric matrices. We realize some of these D-modules explicitly as subquotients in the pole order filtration associated to the determinant/Pfaffian of a generic matrix, and others as local cohomology modules. We give a direct proof of a conjecture of Levasseur in the case of general and skew-symmetric matrices, and provide counterexamples in the case of symmetric matrices. The character calculations are used in subsequent work with Weyman to describe the D-module composition factors of local cohomology modules with determinantal and Pfaffian support.

math.AG↗

arXiv2015

Local cohomology with support in ideals of symmetric minors and Pfaffians

We compute the local cohomology modules H_Y^(X,O_X) in the case when X is the complex vector space of n x n symmetric, respectively skew-symmetric matrices, and Y is the closure of the GL-orbit consisting of matrices of any fixed rank, for the natural action of the general linear group GL on X. We describe the D-module composition factors of the local cohomology modules, and compute their multiplicities explicitly in terms of generalized binomial coefficients. One consequence of our work is a formula for the cohomological dimension of ideals of even minors of a generic symmetric matrix: in the case of odd minors, this was obtained by Barile in the 90s. Another consequence of our work is that we obtain a description of the decomposition into irreducible GL-representations of the local cohomology modules (the analogous problem in the case when X is the vector space of m x n matrices was treated in earlier work of the authors).

math.AC↗

arXiv2015

Characters of equivariant D-modules on Veronese cones

For d > 1, we consider the Veronese map of degree d on a complex vector space W , Ver_d : W -> Sym^d W , w -> w^d , and denote its image by Z. We describe the characters of the simple GL(W)-equivariant holonomic D-modules supported on Z. In the case when d is 2, we obtain a counterexample to a conjecture of Levasseur by exhibiting a GL(W)-equivariant D-module on the Capelli type representation Sym^2 W which contains no SL(W)-invariant sections. We also study the local cohomology modules H_Z^j(S), where S is the ring of polynomial functions on the vector space Sym^d W. We recover a result of Ogus showing that there is only one local cohomology module that is non-zero (namely in degree j = codim(Z)), and moreover we prove that it is a simple D-module and determine its character.

math.AG↗

arXiv2014

Introduction to uniformity in commutative algebra

These notes are based on three lectures given by the first author as part of an introductory workshop at MSRI for the program in Commutative Algebra, 2012-13. The notes follow the talks, but there are extra comments and explanations, as well as a new section on the uniform Artin-Rees theorem. The notes deal with the theme of uniform bounds, both absolute and effective, as well as uniform annihilation of cohomology

math.AC↗

arXiv2013

Local cohomology with support in ideals of maximal minors and sub-maximal Pfaffians

We compute the GL-equivariant description of the local cohomology modules with support in the ideal of maximal minors of a generic matrix, as well as of those with support in the ideal of 2n x 2n Pfaffians of a (2n+1)x(2n+1) generic skew-symmetric matrix. As an application, we characterize the Cohen-Macaulay modules of covariants for the action of the special linear group SL(G) on G^m. The main tool we develop is a method for computing certain Ext modules based on the geometric technique for computing syzygies and on Matlis duality.

math.AC↗

arXiv2013

Local cohomology with support in generic determinantal ideals

For positive integers m >= n >= p, we compute the GL_m x GL_n-equivariant description of the local cohomology modules of the polynomial ring S of functions on the space of m x n matrices, with support in the ideal of p x p minors. Our techniques allow us to explicitly compute all the modules Ext_S(S/I_x,S), for x a partition and I_x the ideal generated by the irreducible sub-representation of S indexed by x. In particular we determine the regularity of the ideals I_x, and we deduce that the only ones admitting a linear free resolution are the powers of the ideal of maximal minors of the generic matrix, as well as the products between such powers and the maximal ideal of S.

math.AG↗

arXiv2013

Non-simplicial decompositions of Betti diagrams of complete intersections

We investigate decompositions of Betti diagrams over a polynomial ring within the framework of Boij--Soederberg theory. That is, given a Betti diagram, we decompose it into pure diagrams. Relaxing the requirement that the degree sequences in such pure diagrams be totally ordered, we are able to define a multiplication law for Betti diagrams that respects the decomposition and allows us to write a simple expression the decomposition of the Betti diagram of any complete intersection in terms of the degrees of its minimal generators. In the more traditional sense, the decomposition of complete intersections of codimension at most 3 are also computed as given by the totally ordered decomposition algorithm obtained from (Eisenbud-Schreyer, 2009). In higher codimension, obstructions arise that inspire our work on an alternative algorithm.

math.AC↗

arXiv2013

3x3 Minors of Catalecticants

Secant varieties to Veronese embeddings of projective space are classical varieties whose equations are not completely understood. Minors of catalecticant matrices furnish some of their equations, and in some situations even generate their ideals. Geramita conjectured that this is the case for the secant line variety of the Veronese variety, namely that its ideal is generated by the 3x3 minors of any of the "middle" catalecticants. Part of this conjecture is the statement that the ideals of 3x3 minors are equal for most catalecticants, and this was known to hold set-theoretically. We prove the equality of 3x3 minors and derive Geramita's conjecture as a consequence of previous work by Kanev.

math.AG↗

arXiv2013

Products of Young symmetrizers and ideals in the generic tensor algebra

We describe a formula for computing the product of the Young symmetrizer of a Young tableau with the Young symmetrizer of a subtableau, generalizing the classical quasi-idempotence of Young symmetrizers. We derive some consequences to the structure of ideals in the generic tensor algebra and its partial symmetrizations. Instances of these generic algebras appear in the work of Sam and Snowden on twisted commutative algebras, as well as in the work of the author on the defining ideals of secant varieties of Segre-Veronese varieties, and in joint work of Oeding and the author on the defining ideals of tangential varieties of Segre-Veronese varieties.

math.CO↗

arXiv2013

Tangential varieties of Segre-Veronese varieties

We determine the minimal generators of the ideal of the tangential variety of a Segre-Veronese variety, as well as the decomposition into irreducible GL-representations of its homogeneous coordinate ring. In the special case of a Segre variety, our results confirm a conjecture of Landsberg and Weyman.

math.AG↗

arXiv2012

Representation stability for syzygies of line bundles on Segre--Veronese varieties

The rational homology groups of the packing complexes are important in algebraic geometry since they control the syzygies of line bundles on projective embeddings of products of projective spaces (Segre--Veronese varieties). These complexes are a common generalization of the multidimensional chessboard complexes and of the matching complexes of complete uniform hypergraphs, whose study has been a topic of interest in combinatorial topology. We prove that the multivariate version of representation stability, a notion recently introduced and studied by Church and Farb, holds for the homology groups of packing complexes. This allows us to deduce stability properties for the syzygies of line bundles on Segre--Veronese varieties. We provide bounds for when stabilization occurs and show that these bounds are sometimes sharp by describing the linear syzygies for a family of line bundles on Segre varieties. As a motivation for our investigation, we show in an appendix that Ein and Lazarsfeld's conjecture on the asymptotic vanishing of syzygies of coherent sheaves on arbitrary projective varieties reduces to the case of line bundles on a product of (at most three) projective spaces.

math.AG↗

arXiv2011

Secant Varieties of Segre--Veronese Varieties

We prove that the ideal of the variety of secant lines to a Segre--Veronese variety is generated in degree three by minors of flattenings. In the special case of a Segre variety this was conjectured by Garcia, Stillman and Sturmfels, inspired by work on algebraic statistics, as well as by Pachter and Sturmfels, inspired by work on phylogenetic inference. In addition, we describe the decomposition of the coordinate ring of the secant line variety of a Segre--Veronese variety into a sum of irreducible representations under the natural action of a product of general linear groups.

math.AG↗

arXiv2009

Affine Toric Equivalence Relations are Effective

Any map of schemes $X\to Y$ defines an equivalence relation $R=X\times_Y X\to X\times X$, the relation of "being in the same fiber". We have shown elsewhere that not every equivalence relation has this form, even if it is assumed to be finite. By contrast, we prove here that every toric equivalence relation on an affine toric variety does come from a morphism and that quotients by finite toric equivalence relations always exist in the affine case. In special cases, this result is a consequence of the vanishing of the first cohomology group in the Amitsur complex associated to a toric map of toric algebras. We prove more generally the exactness of the Amitsur complex for maps of commutative monoid rings.

math.AG↗