Abstracts

Torus invariants and binomial D-modules

Abstract: Binomial D-modules are given by a binomial ideal and homogeneity operators. Combinatorial tools from toric geometry have been successful at analyzing many aspects of binomial D-modules, which carry a torus action. We will consider how to interpret taking invariants of D-modules with torus actions, with the goal of gaining a new understanding of the classical hypergeometric systems studied by, among others, Gauss, Appell, and Lauricella. This is joint work with Laura Felicia Matusevich and Uli Walther.

Transcendence of the Field of Periods for Special Cubic Threefolds

Abstract. We define the field of periods of an algebraic variety, discuss its transcendence degree and its relation to the Mumford-Tate group, then do some computations for a special class of cubic threefolds.

Lex-Plus-Power inequalities

Abstract: I will discuss a conjecture by Eisenbud, Green and Harris regarding the possible Hilbert functions of subschemes of complete intersections. Two special cases of this conjecture are the Cayley-Bacharach theorem and the Kruskal-Katona theorem on f-vectors of abstract simplicial complexes. I will focus on a series of sharp upper bounds for multiplicity, Betti numbers and Hilbert functions of local cohomology modules that are suggested by, and can be proved when the above conjecture is known.

Gromov-Witten Theory of Calabi-Yau Three-Orbifolds

Abstract: Two fundamental questions in Gromov-Witten theory are Ruan's crepant resolution conjecture and the Gromov-Witten/Donaldson-Thomas correspondence. I will introduce these notions and discuss recent results leading to a complete understanding of the correspondences when the targets belong to a certain class of toric orbifold. This is joint work with A. Brini, R. Cavalieri, and Z. Zong.

F-singularities vs singularities of higher dimensional algebraic geometry

Abstract: In this talk, I will motivate the study of F-singularities, the singularities defined by Frobenius, via connection with Kodaira type vanishing theorems. We will discuss the close relationship between F-singularities and singularities like rational and log canonical singularities. Finally some recent joint work Yoshinori Gongyo and Paolo Cascini will be discussed showing uniform bounds for the equality of F-regular and log terminal singularities on surfaces of characteristic p.

Deformations of Minimal Cohomology Classes

Abstract: We study infinitesimal deformations of the Brill-Noether locus W_d(C) parameterizing degree d line bundles on a smooth curve C, and of the inclusion W_d(C) ⊂ J(C) of W_d(C) in its Albanese variety, the Jacobian variety J(C) of C. We will show that if there is a deformation of W_d(C) which is not induced by its resolution of singularities C_d, the d-symmetric product of C, or if there is a deformation of W_d(C) that deforms J(C) in a direction which goes outside the Jacobian locus, then C is hyperelliptic. This is joint work with L. Lombardi