Bring your lunch to the Vincent Hall courtyard at 12:15pm to catch up with (or meet for the first time) others in the algebra group.
Talk Abstract:
Building on a result of Swanson, Cutkosky--Herzog--Trung and Kodiyalam described the surprisingly predictable asymptotic behavior of Castelnuovo--Mumford regularity for powers of ideals on a projective space P^n: given an ideal I, there exist integers d and e such that for the regularity of I^n satisfies dn+e.
Through a medley of examples we will see why asking the same question about an ideal I in the total coordinate ring S of a smooth projective toric variety X is interesting. After that I will summarize the ideas and methods we used to bound the region reg(I^n) a subset of Pic(X) by proving that it contains a translate of reg(S) and is contained in a translate of Nef(X), with each bound translating by a fixed vector as n increases. Along the way will see some surprising behavior for multigraded regularity of modules.
We construct explicit GL-equivariant minimal free resolutions of certain (truncations of) modules of relative invariants over Veronese subrings in arbitrary characteristic. The free modules in the resolution correspond to certain skew Schur modules associated to "ribbon" or "skew-hook" diagrams, and the differentials at each step are surprisingly uniform. We then utilize the uniformity of these resolutions to explicitly compute information about Tor and Hom between these modules and show that they also have rather simple descriptions in terms of ribbon skew-Schur modules. This is joint work with Michael Perlman, Sasha Pevzner, Vic Reiner, and Keller Vandebogert.
Given a closed subvariety Z in a smooth complex variety X, the local cohomology sheaves with support in Z are holonomic D-modules, and thus have finite filtration with simple composition factors. We will discuss the D-module structure on local cohomology in the case when X is a Grassmannian and Z is a Schubert variety, including a combinatorial formula describing the composition factors and the weight filtration in the sense of mixed Hodge modules. Upon restriction to the "opposite big cell", these calculations recover several previously known results concerning local cohomology with support in determinantal varieties.
Boij-Söderberg theory gives a combinatorial description of the set of Betti tables belonging to finite length modules over the polynomial ring. In this talk, I'll discuss a conjectured generalization of Boij-Söderberg theory to a more general class of objects called differential modules. I'll give an introduction to differential modules as natural generalizations of chain complexes and talk about some recent results towards proving a differential module version of the Boij-Söderberg conjectures.
Let a finite group G act on the polynomial ring S = k[x_1,...,x_n] via graded ring automorphisms. The ring of invariants S^G is the largest k-submodule of S on which G acts trivially, and it is a long-studied object in commutative algebra. Instead, we consider the cofixed space S_G, which is the largest k-module quotient of S on which G acts trivially. While S_G is not a ring, it carries the structure of a module over S^G. Working integrally and setting G to be the symmetric group on n letters, we can embed the cofixed space as an ideal inside the ring of symmetric polynomials. Doing so gives rise to a family of ideals - one for each n. Localizing the coefficient ring of S at a prime p reveals striking behavior in these ideals, which stay stable (in a sense) as n grows, but jump in complexity each time n equals a multiple of p. In this talk, we will discuss the construction of this family of ideals, as well as some results and conjectures on its structure.
Recently, W. Zhang has posted a question about asymptotic behavior of socle degrees of local cohomology modules. In this talk we will discuss this question in the following setting: let S be either Sym(C^m \otimes C^n) or Sym(\bigwedge^2 C^{2n+1}) and let I be either the determinantal ideal of maximal minors or the sub-maximal Pfaffians, respectively. We will show that the socle of H^j_\frakm (S/I^t) is generated in one degree for t \geq n and answer the question of Zhang in our setting. In particular, given the GL-decomposition of Ext^{mn-j}_S(S/I^t, S) by Raicu, Weyman and Witt, we will recover its S-module structure. This is work in progress with Michael Perlman.
Hilbert revolutionized proving finite generation statements with his famous Hilbert's basis theorem. Since then, Noetherianity has become a well-studied and, for many objects of interest, relatively well understood concept in commutative algebra. The same cannot be said for some of the most basic objects in equivariant commutative algebra.
We will begin this talk by defining Noetherianity in this equivariant setting, and then discuss largely open Noetherianity problems for some of the most basic objects: twisted commutative algebras. We will then consider an approach to Noetherianity using initial ideals, a counterexample to such an approach, and some recent progress stemming from this counterexample.
I will present the calculation of the rational Borel-Moore homology groups for affine determinantal varieties, thus solving a problem of Pragacz and Ratajski. The main ingredients are the relation with Hartshorne's algebraic de Rham homology theory, the calculation of the singular cohomology of matrix orbits, and the degeneration of the Cech-de Rham spectral sequence. As a consequence, we obtain explicit formulas for the dimensions of de Rham cohomology groups of local cohomology with determinantal support, which are analogues of Lyubeznik numbers first introduced by Switala. We further determine the Hodge numbers of the singular cohomology of matrix orbits and of the Borel–Moore homology of their closures, based on Saito’s theory of mixed Hodge modules and a recent result of Perlman. This is joint work with Claudiu Raicu.
I will consider a class of algebras, “Homotopy Path Algebras”, naturally appearing in many contexts; e.g. algebraic topology, sheaf theory, and toric geometry (as full strong exceptional collections of line bundles). I will develop the general theory of such algebras and briefly explain the connection to homological mirror symmetry, expanding on the ideas of Bondal and Fan-Lui-Truemann-Zaslow. This is based on joint work with Jesse Huang.
This is joint work with Mahrud Sayrafi. Given a variety X as in the title, I will explain a construction of a resolution of the diagonal for X that has the shortest possible length and whose terms are sums of line bundles. Applications include (1) a new case of a conjecture of Berkesch-Erman-Smith that predicts a version of Hilbert’s Syzygy Theorem for virtual resolutions, (2) a Horrocks-type splitting criterion for vector bundles over smooth projective toric varieties of Picard rank 2, extending a result of Eisenbud-Erman-Schreyer, and (3) a new proof of a special case of a conjecture of Orlov on the Rouquier dimension of derived categories.
A Landau-Ginzburg (LG) model is a triplet of data (X, W, G) consisting of a regular function W from a quasi-projective variety X with a group G acting on X leaving W invariant. One can build an analogue of Hodge theory and period integrals associated to an LG model when G is trivial. This involves oscillatory integrals on certain cycles in X (fear not: this is actually cute and will be done in examples!). Mirror symmetry states that period integrals often encode enumerative geometry and this is also the case here. An enumerative theory developed by Fan, Jarvis, and Ruan gives FJRW invariants, the analogue of Gromov-Witten invariants for LG models. These invariants are now called FJRW invariants. A problem is that finding the right deformation period integrals is hard. We define and use a new open enumerative theory for certain Landau-Ginzburg LG models to solve this problem in low dimension.
Roughly speaking, this involves computing specific integrals on certain moduli of disks with boundary and interior marked points. One can then construct a mirror Landau-Ginzburg model to a Landau-Ginzburg model using these invariants that gives you the right deformation for free. This allows us to prove a mirror symmetry result analogous to that established by Cho-Oh, Fukaya-Oh-Ohta-Ono, and Gross for mirror symmetry for toric Fano manifolds. This is joint work with Mark Gross and Ran Tessler.
The theory of F-modules, pioneered by Lyubeznik, is a powerful machinery that allows us to prove finiteness results about local cohomology of regular rings in positive characteristic. In this talk I will explain how this theory can be extended to rings with mild singularities (namely: rings with finite F-representation type). I will then show how one can recover and extend some results on local cohomology for these rings.