Commutative Algebra Seminar Spring 2019

Commutative Algebra Seminar meets on Thursday 1:25-2:15 in Vincent 301.


Date Speaker Title
February 28 Jay Yang (local) Virtual Resolutions of Monomial Ideals
March 14 Aleksandra Sobieska-Snyder (Texas A&M) Toward Free Resolutions Over Scrolls
March 28 Thomas Bitoun (University of Toronto) Bernstein-Sato polynomials in positive characteristic and Hodge theory
April 4 Zhang Wenliang (University of Illinois at Chicago) An Analogue of the Hartshorne-Polini Theorem in Positive Characteristic
April 11 Juliette Bruce (University of Wisconsin) Semi-Ample Asymptotic Syzygies


Virtual Resolutions of Monomial Ideals

Virtual resolutions as defined by Berkesch, Erman, and Smith, provide a more geometrically meaningful generalization of free resolutions in the case of subvarieties of a toric variety. In this setting I prove an analog of Hilbert's syzygy theorem for virtual resolutions of monomial ideals in toric varieties subject to some mild conditions.

Towards Free Resolutions Over Scrolls

Free resolutions over the polynomial ring have a storied and active record of study. However, resolutions over other rings are much more mysterious; even simple examples can be infinite! In these cases, we look to any combinatorics of the ring to glean information. This talk will present a minimal free resolution of the ground field over the semigroup ring arising from rational normal 2-scrolls, and (if time permits) a computation of the Betti numbers of the ground field for all rational normal k-scrolls.

Bernstein-Sato polynomials in positive characteristic and Hodge theory

Bernstein-Sato polynomials are fundamental in D-module theory. For example, they are the main finiteness ingredient in the construction of nearby cycles. We will present a positive characteristic analogue of the Bernstein-Sato polynomials. After diving in the world of characteristic p D-modules, we shall consider how our construction varies with the prime p. This turns out to be related to questions of Hodge theory and Poisson homology.

An Analogue of the Hartshorne-Polini Theorem in Positive Characteristic

Recently, Hartshorne and Polini proved a theorem to characterize the dimension of certain de Rham cohomology groups of a holonomic D-module over complex numbers as the number of specific D-linear maps associated with the holonomic D-module. In this talk, I will explain an analogue of this result in positive characteristic for F-finite F-modules. This is a joint work with Nicholas Switala.

Semi-Ample Asymptotic Syzygies

I will discuss the asymptotic non-vanishing of syzygies for products of projective spaces, generalizing the monomial methods of Ein-Erman-Lazarsfeld. This provides the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semi-ample fashion behave in nuanced and previously unseen ways.

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