Claudiu Raicu
Department of Mathematics
University of Notre Dame
255 Hurley
Notre Dame, IN 46556
I am an Associate Professor in the math department at University of Notre Dame. Previously I was an Instructor in the math department at Princeton University. During the Spring 2013 semester I was a post-doctoral fellow at MSRI. I received my Ph.D. in May 2011 from the math department at UC Berkeley. My advisor was David Eisenbud. My thesis was titled Secant Varieties of Segre-Veronese Varieties. I am interested in algebraic geometry, commutative algebra and their computational aspects. Here is my CV.
Papers
- Koszul modules and Greenâ€™s conjecture, with Marian Aprodu, Gavril Farkas, Ştefan Papadima and Jerzy Weyman (Invent. Math, or arXiv: 1810.11635)
We prove a strong vanishing result for finite length Koszul modules, and use it to derive Green's conjecture for every g-cuspidal rational curve over an algebraically closed field k with char(k) = 0 or char(k) >= (g+2)/2. As a consequence, we deduce that the general canonical curve of genus g satisfies Green's conjecture in this range. Our results are new in positive characteristic, whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by Voisin. Our strategy involves establishing two key results of independent interest: (1) we describe an explicit, characteristic-independent version of Hermite reciprocity for sl_2-representations; (2) we completely characterize, in arbitrary characteristics, the (non-)vanishing behavior of the syzygies of the tangential variety to a rational normal curve.
- Syzygies of determinantal thickenings and representations of the general linear Lie superalgebra, with Jerzy Weyman (Acta Math. Vietnam. 44, no. 1, 269—284, 2019, or arXiv: 1808.05649)
We let S denote the ring of polynomial functions on the space of m x n matrices, and consider the action of the group GL = GL_m x GL_n via row and column operations on the matrix entries. For a GL-invariant ideal I in S we show that the linear strands of its minimal free resolution translate via the BGG correspondence to modules over the general linear Lie superalgebra gl(m|n). When I is the ideal generated by the GL-orbit of a highest weight vector, we give a conjectural description of the classes of these gl(m|n)-modules in the Grothendieck group, and prove that our prediction is correct for the first strand of the minimal free resolution.
- Topological invariants of groups and Koszul modules, with Marian Aprodu, Gavril Farkas, Ştefan Papadima and Jerzy Weyman (arXiv: 1806.01702)
We provide a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace K inside the second exterior product of a vector space, as well as a sharp upper bound for its Hilbert function. This purely algebraic statement has interesting applications to the study of a number of invariants associated to finitely generated groups, such as the Alexander invariants, the Chen ranks, or the degree of growth and nilpotency class. For instance, we explicitly bound the aforementioned invariants in terms of the first Betti number for the maximal metabelian quotients of (1) the Torelli group associated to the moduli space of curves; (2) nilpotent fundamental groups of compact Kaehler manifolds; (3) the Torelli group of a free group.
- Iterated local cohomology groups and Lyubeznik numbers for determinantal rings, with András C. Lőrincz (arXiv: 1805.08895)
We give an explicit recipe for determining iterated local cohomology groups with support in ideals of minors of a generic matrix in characteristic zero, expressing them as direct sums of indecomposable D-modules. For non-square matrices these indecomposables are simple, but this is no longer true for square matrices where the relevant indecomposables arise from the pole order filtration associated with the determinant hypersurface. Specializing our results to a single iteration, we determine the Lyubeznik numbers for all generic determinantal rings, thus answering a question of Hochster.
- Equivariant D-modules on binary cubic forms, with András C. Lőrincz and Jerzy Weyman (Comm. Algebra 47, no. 6, 2457—2487, 2019, or arXiv: 1712.09932)
We consider the space X = Sym^3(C^2) of binary cubic forms, equipped with the natural action of the group GL_2 of invertible linear transformations of C^2. We describe explicitly the category of GL_2-equivariant coherent D_X-modules as the category of representations of a quiver with relations. We show moreover that this quiver is of tame representation type and we classify its indecomposable representations. We also give a construction of the simple equivariant D_X-modules (of which there are 14), and give formulas for the characters of their underlying GL_2-representations. We conclude the article with an explicit calculation of (iterated) local cohomology groups with supports given by orbit closures.
- Homological invariants of determinantal thickenings
(Bull. Math. Soc. Sci. Math. Roumanie, Volume 60 (108), no. 4:425—446, 2017, or arXiv: 1712.09938)
The study of homological invariants such as Tor, Ext and local cohomology modules constitutes an important direction in commutative algebra. Explicit descriptions of these invariants are notoriously difficult to find and often involve combining an array of techniques from other fields of mathematics. In recent years tools from algebraic geometry and representation theory have been successfully employed in order to shed some light on the structure of homological invariants associated with determinantal rings. The goal of this notes is to survey some of these results, focusing on examples in an attempt to clarify some of the more technical statements.
- On the (non-)vanishing of syzygies of Segre embeddings, with Luke Oeding and Steven V Sam (arXiv: 1708.03803)
We analyze the vanishing and non-vanishing behavior of the graded Betti numbers for Segre embeddings of products of projective spaces. We give lower bounds for when each of the rows of the Betti table becomes non-zero, and prove that our bounds are tight for Segre embeddings of products of P^1. This generalizes results of Rubei concerning the Green-Lazarsfeld property N_p for Segre embeddings. Our methods combine the Kempf-Weyman geometric technique for computing syzygies, the Ein-Erman-Lazarsfeld approach to proving non-vanishing of Betti numbers, and the theory of algebras with straightening laws.
- Regularity and cohomology of determinantal thickenings
(Proceedings of the London Mathematical Society (3) 116, no. 2:248—280, 2018, or arXiv: 1611.00415)
We consider the ring S=C[x_ij] of polynomial functions on the vector space C^(m x n) of complex m x n matrices. We let GL= GL_m x GL_n and consider its action via row and column operations on C^(m x n) (and the induced action on S). For every GL-invariant ideal I in S and every j>=0, we describe the decomposition of the modules Ext^j_S(S/I,S) into irreducible GL-representations. For any inclusion I into J of GL-invariant ideals we determine the kernels and cokernels of the induced maps Ext^j_S(S/I,S) -> Ext^j_S(S/J,S). As a consequence of our work, we give a formula for the regularity of the powers and symbolic powers of generic determinantal ideals, and in particular we determine which powers have a linear minimal free resolution. As another consequence, we characterize the GL-invariant ideals I in S for which the induced maps Ext^j_S(S/I,S) -> H_I^j(S) are injective. In a different direction we verify that Kodaira vanishing, as described in work of Bhatt-Blickle-Lyubeznik-Singh-Zhang, holds for determinantal thickenings.
- Bernstein—Sato polynomials for maximal minors and sub-maximal Pfaffians, with András C. Lőrincz, Uli Walther, and Jerzy Weyman (Advances in Mathematics 307:224—252, 2017, or arXiv: 1601.06688)
We determine the Bernstein-Sato polynomials for the ideal of maximal minors of a generic m x n matrix, as well as for that of sub-maximal Pfaffians of a generic skew-symmetric matrix of odd size. As a corollary, we obtain that the Strong Monodromy Conjecture holds in these two cases.
- Local cohomology with support in ideals of symmetric minors and Pfaffians, with Jerzy Weyman (Journal of the London Mathematical Society (2) 94, no. 3:709—725, 2016, or arXiv: 1509.03954)
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. 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.
- Characters of equivariant D-modules on spaces of matrices
(Compositio Mathematica, Volume 152, Issue 9:1935—1965, 2016, or arXiv: 1507.06621)
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. In subsequent joint work with Weyman, the character calculations will be used to describe the local cohomology modules with determinantal and Pfaffian support.
- Characters of equivariant D-modules on Veronese cones
(Transactions of the American Mathematical Society 369, no. 3: 2087—2108, 2017, or arXiv: 1412.8148)
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.
- The syzygies of some thickenings of determinantal varieties, with Jerzy Weyman (Proceedings of the American Mathematical Society 145:49—59, 2017, or arXiv: 1411.0151)
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.
- Representation stability for syzygies of line bundles on Segre-Veronese varieties,
(Journal of the
European Mathematical Society 18, no. 6:1201—1231, 2016, or arXiv: 1209.1183)
We prove that the multivariate version of the notion of representation stability introduced by Church and Farb holds for the syzygies of line bundles on Segre-Veronese varieties. We give bounds for when stabilization occurs and show that they are sometimes sharp by describing the linear syzygies for a family of line bundles on Segre varieties. To motivate our work, we show in an appendix that Ein and Lazarsfeld's conjecture on the asymptotic vanishing of syzygies of arbitrary varieties reduces to the case of a product of three projective spaces.
- Introduction to uniformity in commutative algebra, with Craig Huneke (Mathematical Sciences Research Institute Publications 67:163—190, Commutative Algebra and Noncommutative Algebraic
Geometry, Cambridge Univ. Press, Cambridge, 2015, or arXiv: 1408.7098)
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.
- Non-simplicial decompositions of Betti diagrams of complete intersections, with Courtney Gibbons, Jack Jeffries, Sarah Mayes, Branden Stone, and Bryan White
(Journal of Commutative Algebra 7, no. 2, 189-206, 2015, or
arXiv: 1301.3441)
We investigate decompositions of Betti diagrams into pure diagrams, in the framework of Boij-Söderberg theory over a standard graded polynomial ring. Relaxing the requirement that the degree sequences in such pure diagrams be totally ordered, we are able to obtain a simple expression of the decomposition of the Betti diagram of any complete intersection in terms of the degrees of its minimal generators.
- Local cohomology with support in generic determinantal ideals, with Jerzy Weyman (Algebra & Number Theory 8, no. 5: 1231-1257, 2014, or arXiv: 1309.0617)
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, and in particular determine the regularity of the ideals I_x.
- Local cohomology with support in ideals of maximal minors and sub-maximal Pfaffians, with Jerzy Weyman and Emily Witt (Advances in Mathematics 250: 596-610, 2014, or arXiv: 1305.1719)
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 on a direct sum of copies of the standard representation.
- Products of Young symmetrizers and ideals in the generic tensor algebra
(Journal of Algebraic Combinatorics 39, no. 2: 247-270, 2014, or
arXiv: 1301.7511)
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. These consequences are essential in the proofs of the main results in Tangential Varieties of Segre-Veronese Varieties.
- Tangential Varieties of Segre-Veronese Varieties, with Luke Oeding,
(Collectanea Mathematica 65, no. 3:303-330, 2014, or
arXiv: 1111.6202)
We build on the techniques from Secant Varieties of Segre-Veronese Varieties and Products of Young symmetrizers and ideals in the generic tensor algebra, obtaining the description of the minimal generators of the ideal of the tangential variety of an arbitrary Segre-Veronese variety. In the special case of a Segre variety, this confirms a conjecture of Landsberg and Weyman.
- 3x3 Minors of Catalecticants,
(Mathematical Research Letters 20, no. 4: 745-756, 2013, or
arXiv: 1011.1564)
We answer a question of Geramita, by proving the equality of the ideals of 3x3 minors of the ``middle'' catalecticant matrices. This was predicted based on Macaulay's theorem on the growth of the Hilbert function of an Artin algebra, and was known to hold for the special case of Hankel matrices. The techniques we introduce give a new perspective on the problem of determining the ideals of secant varieties to Veronese varieties.
- Secant Varieties of Segre-Veronese Varieties,
(Algebra & Number Theory 6, no. 8: 1817-1868, 2012, or
arXiv: 1011.5867)
We prove a generalized version of a conjecture of Garcia, Stillman and Sturmfels, stating that the ideal of the first secant variety of a Segre-Veronese embedding of a product of projective spaces is generated by 3x3 minors of flattenings. We introduce techniques similar to the ones in 3x3 Minors of Catalecticants, providing a new strategy for analyzing the ideals and coordinate rings of secant varieties to Segre-Veronese varieties.
- Affine Toric Equivalence Relations are Effective,
(Proceedings of the American Mathematical Society 138: 3835-3847, 2010, or
arXiv: 0905.4805)
We prove that if X is an affine toric variety, and R is an equivalence relation on X, preserved by the diagonal torus action, then R comes from a toric map X->Y. Moreover, if R is finite then there exists a geometric quotient X/R. We also show that the Amitsur complex associated to a map of monoid rings (defined at the monoid level) is exact. This gives a new class of ring extensions, besides the augmented and faithfully flat ones, for which the Amitsur complex has no nontrivial cohomology.
- Appendix to János Kollár, Quotients by Finite Equivalence Relations,
(Mathematical Sciences Research Institute Publications, 59, Current developments in algebraic geometry, 227-256, Cambridge Univ. Press, Cambridge, 2012, or arXiv: 0812.3608)
We answer a question of Kollár, by providing an example of a noneffective finite equivalence relation R on a two-dimensional affine space X. In other words, R is not constructed as the fiber product of X with itself over a finite base. We construct our example by employing an interesting relationship between equivalence relations and the cohomology of Amitsur complexes.
M2 code that verifies the assertions made in this appendix.
Reading Seminars
Macaulay2
With Michael K. Brown, Hang Huang, Robert P. Laudone, Michael Perlman, Steven V Sam, Joao Pedro Santos, we wrote the package SchurComplexes for computing Schur complexes. A short presentation of the package is available here.
With Mike Perlman, we wrote the package GLmnReps for computing with representations of the Lie superalgebra gl(m|n).
Here is the most up to date version (with documentation) of the Macaulay2 SchurRings package, developed jointly with Mike Stillman. Note that this code is not (yet) incorporated in Macaulay2. A short presentation of the package is available here.
I also wrote a package implementing the push forward functor for finite ring maps.
Posters and Slides
- Slides on local cohomology with support in determinantal ideals, from the SIAM Conference on Applied Algebraic Geometry, August 1-4, 2013 at Colorado State University, Fort Collins.
- Slides on tangential varieties of Segre-Veronese varieties, from the SIAM Conference on Applied Algebraic Geometry, August 1-4, 2013 at Colorado State University, Fort Collins.
- Slides on local cohomology modules with support in ideals of maximal minors and sub-maximal Pfaffians, from the Joint International Meeting of the American Mathematical Society and the Romanian Mathematical Society, June 27 - 30, 2013, Alba Iulia, Romania.
- Slides on secant varieties of Segre-Veronese varieties, from the SIAM Conference on Applied Algebraic Geometry, October 6-9, 2011 at North Carolina State University, Raleigh.
- Poster on the GSS conjecture, from WAGS, Nov 6-7, 2010 at University of Arizona, Tucson.
- Slides on finite equivalence relations, from the Eighth AMS-SMM International Meeting, June 4, 2010 at UC Berkeley.
- Poster on finite equivalence relations, from WAGS, May 1-2, 2010 at UBC, Vancouver.
Other writings
Notes compiled together with Enric Nart during the ``Computing Integral Closure'' Workshop at MSRI, July 2010.
Mathematics Research Communities report on Boij-Söderberg Theory in the nonstandard graded case, with B. Barwick, J. Biermann, D. Cook II, W. F. Moore and D. Stamate, Snowbird, June 2010.
Teaching
Spring 2019 :
Math 40510 - Introduction to Algebraic Geometry
Math 80210 - Homological Commutative Algebra and Symmetry
Fall 2018 :
Math 20550 - Multivariable Calculus
Spring 2018 :
Math 30820 - Honors Algebra IV
Fall 2017 :
Math 20580 - Introduction to Linear Algebra and Differential Equations
Math 30810 - Honors Algebra III
Spring 2017 :
Math 80620 - Algebraic Geometry II
Fall 2016 :
Math 60710 - Algebraic Geometry I
Spring 2016 :
Math 80220 - Introduction to Toric Varieties
Fall 2015 :
Math 10560 - Calculus II
Spring 2015 :
Math 20580 - Introduction to Linear Algebra and Differential Equations
Spring 2014 :
Math 346 - Algebra II
Fall 2013 :
Math 345 - Algebra I
Math 984 - Junior seminar on Coxeter groups
Fall 2012 :
Math 345 - Algebra I
Spring 2012 :
Math 202 - Linear Algebra
Fall 2011 :
Math 201 - Multivariable Calculus
Spring 2011 :
Math 16B - Analytic Geometry and Calculus
Fall 2010 :
Math 54 - Linear Algebra
Spring 2010 :
Math 1B - Calculus
Fall 2009 :
Math 1B - Calculus