Commutative Algebra -- Algebraic Geometry in the Southeast
Columbia, SC
November 8 --10, 2013.
Abstracts:
- Speaker: Nicolas Addington (Duke University)
Title: On derived categories of moduli spaces of torsion sheaves on K3
Abstract: If S is a K3 surface and M a moduli space of certain torsion sheaves on S, then using Arinkin's work on compactified Jacobians I can show that the functor F from the derived category of S to that of M induced by the universal sheaf is a so-called "P^n-functor." This has two interesting consequences: (1) It yields autoequivalences of D(M), which should be seen as "hidden symmetries" of M. (2) As one deforms M out into the wider world of hyperkaehler manifold, S does not deform along with it, but the functor F may allow one to deform D(S), and the Mukai lattice of S (a slight enlargement of the Hodge structure on H^2(S)), along with M. Conjecturally, F is a P^n-functor when M is any moduli space of sheaves on S; in earlier work I showed this for M = Hilb^{n+1}(S), where the geometry is quite different. This is joint work with Ciaran Meachan.
- Speaker: Neil Epstein (George Mason University)
Title: "The tight interior of a module: a dual to tight closure"
(joint with Karl Schwede) Click here for more details.
- Speaker: Angela Gibney (University of Georgia)
Title: Conformal blocks and quantum cohomology
Abstract: I will talk about recent work with Belkale and Mukhopadyay, in which we study aspects of vector bundles of conformal blocks on the moduli space of curves using the quantum cohomology of Grassmannians. In particular, we show that above the critical level, which we introduce, all vector bundles of type A conformal blocks on $\overline{{M}}_{0,n}$ are trivial. We uncover new level-rank symmetries between pairs of critical level conformal blocks divisors, and applications about maps they define. I'll explain how these results might be interpreted with respect to existing conjectures about the birational geometry of the moduli space of curves.
Click here and here for more details.
- Speaker: Robin Hartshorne (University of California -- Berkeley)
Title: Set-theoretic complete intersections and divisor class groups.
Abstract:
I will begin by reviewing some of the history of the problem of set-theoretic complete intersections in projective space. Then I will report on some recent work, joint with Claudia Polini, inspired by this problem, which provides a small contribution to the problem itself.
Click here for more details.
- Speaker: Craig Huneke (University of Virginia)
Title: Powers and symbolic powers define Golod rings.
Abstract: This is joint work with Juergen Herzog. Serre gave an upper
bound, term by term, on the Poincare series of a local Noetherian ring,
which is the generating function of the Betti numbers of the residue field.
When the bound is obtained, the ring is said to be Golod. This talk will
discuss some of the history of this idea, then present my recent work
with Herzog proving that powers and symbolic powers of graded ideals
always define Golod rings. We will also touch on some open problems.
Click here for more details.
- Speaker: Ezra Miller (Duke University)
Title: Binomial irreducible decomposition
Abstract: This talk presents a response to Problem 7.5 in the
paper Binomial ideals, by Eisenbud and Sturmfels: "Does
every binomial ideal have an irreducible decomposition into
binomial ideals? Find a combinatorial characterization of
irreducible binomial ideals." Joint work with Thomas Kahle
and Chris O'Neill.
Click here for more details.
- Speaker: Luis Núñez-Betancourt (University of Virginia)
Title: Lyubeznik numbers and injective dimension of local cohomology in mixed characteristic.
Abstract: The Lyubeznik numbers in mixed characteristic are invariants for local rings that do not contain a field. These invariants are inspired by the numbers that Lyubeznik defined for equal characteristic rings, which are known to have algebraic and geometric interpretations. In this talk, we will give an overview of Lyubeznik numbers and present several properties for the invariants in mixed characteristic. We will also discuss related results on injective dimension of local cohomology over regular rings of mixed characteristic. In particular, we will present an example of a local cohomology module whose injective dimension is bigger than the dimension of its support. This differs from the behavior of local cohomology modules in equal characteristic and answers a question raised by Lyubeznik. This is joint work with Daniel Hernández, Felipe Pérez and Emily Witt. Click here for more details.
- Speaker: Joseph Rabinoff (Georgia Institute of Technology)
Title: Continuous analogues of methods used to calculate component groups of Jacobians
Abstract: Let K be complete, discretely-valued field and let X be a smooth
projective K-curve equipped with a semistable model over the valuation
ring. A series of classical theorems, mostly due to Raynaud, give two
ways of calculating the component group of the Jacobian J of X: one
using the intersection matrix on the special fiber of the model of X,
and the other using cycles on its incidence graph G. These
calculations can be interpreted in terms of divisors on G (in the
sense of Baker-Norine) and the uniformization theory of G,
respectively. If K is complete and non-Archimedean but not discretely
valued, these theorems are no longer applicable, as Néron models do
not exist in this situation. Replacing the component group with the
skeleton of J (in the sense of Berkovich), a principally polarized
real torus canonically associated to J, and the incidence graph with a
skeleton Gamma of X, a metric graph, we will prove "continuous"
analogues of these theorems. Specifically, we will show that the
Jacobian of Gamma is canonically identified with the skeleton of J as
principally polarized real tori, in a way that is compatible with the
descriptions of the two Jacobians in terms of divisors and in terms of
uniformizations. As a consequence, we will show that, when K is
algebraically closed, essentially any piecewise-linear function on
Gamma is the restriction to Gamma of -log |f|, where f is a nonzero
rational function on X.
This work is joint with Matt Baker. Click here for more details.
- Speaker: Kirsten Wickelgren (Georgia Institute of Technology)
Title: An Abel map to the compactified Picard scheme realizes Poincaré duality
Abstract: The Picard scheme of an algebraic curve modulo its boundary is compactified to a moduli space of rank 1, torsion free sheaves Picbar, and there is an Abel map X/ (boundary X) -> Picbar. We show that applying étale H_1 to this Abel map realizes Poincaré duality in the sense that there is a canonical isomorphism H_1(Picbar,Z/l) = H^1(X/(boundary X), Z/l(1)), and the composition H_1(X/boundary X,Z/l) -> H_1(Picbar, Z/l) -> H^1(X/boundary X, Z/l(1)) is Poincaré duality. In the process, we construct a Mayer-Vietoris sequence for certain push-outs of schemes along closed immersions, and an isomorphism of functors pi_1^l Pic(-) = H^1(-,Z_l(1)). This is joint work with Jesse Kass.
Click here for more details.
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