Algebraic Geometry/Commutative Algebra Seminar, 2023–2024

To volunteer to give a talk, or for anything else regarding the seminar, contact Claudiu Raicu or Eric Riedl.

Fall Schedule

The seminar will meet on Thursdays, 3:30–4:30pm, either on Zoom or in 258 Hurley, unless otherwise noted. Related events are also listed below.

Abstracts

Aug. 31, 2023

Speaker

Alessandra Costantini (Oklahoma State)

Title

Rees algebras of linearly presented ideals

Abstract

The Rees algebra of an ideal I is an invaluable tool in the study of the algebraic properties of I, as it encodes information on the asymptotic growth of the powers of I. Moreover, as Proj(R(I)) is the blowup of an affine scheme along V(I), Rees algebras represent an essential tool in the study of singularities. As the blowup construction describes Proj(R(I)) via parametric equations, a fundamental problem is to find the implicit equations of blowups. This is a difficult problem in general, as a priori one would need to determine all possible relations among the generators of all powers of I. In this talk, I will restrict to the case when I is a codimension-two perfect ideal in a polynomial ring k[x_1,...,x_d], admitting a presentation matrix consisting of linear entries. Most of the existing literature in this setting assumes the so-called G_d condition that the Fitting ideals Fitt_i(I) have codimension at least i+1 for i=1,...,d-1. Moving away from this assumption, we determine the defining ideal of the Rees algebra of I by requiring only that this codimension constraint is satisfied for i=1,...,d-2. This is part of joint work with Edward Price and Matthew Weaver.

Sep. 7, 2023

Speaker

Keller VandeBogert (Notre Dame)

Title

The Total Rank Conjecture in Characteristic Two

Abstract

The total rank conjecture is a coarser version of the Buchsbaum-Eisenbud-Horrocks conjecture which, loosely stated, predicts that modules with large annihilators must also have "large" syzygies. In 2017, Walker proved that the total rank conjecture holds over rings of odd characteristic, using techniques that heavily relied on the invertibility of 2. In this talk, I will speak on joint work with Mark Walker where we settle (and generalize) the total rank conjecture over k-algebras of arbitrary characteristic. Our techniques take advantage of the classical Dold-Kan correspondence and allow us to prove an even stronger version of the total rank conjecture when k has characteristic 2.

Sep. 14, 2023

Speaker

Anand Patel (Oklahoma State)

Title

Counting cubic surfaces

Abstract

The classical Thom-Porteous formulas essentially tell us how many times a matrix of a particular rank arises in a family of matrices. In other words, it tells us how many times a matrix is equivalent to the particular rank $r$ matrix of the form $\begin{pmatrix}1 &0 &\dots& 0\\ 0 & 1 & \dots & 0\\ \vdots & \vdots & \ddots & 0 \end{pmatrix}$ after performing row and column operations. The formulas themselves are certain expressions in chern classes of the vector bundles involved. The main point of my talk is to advertise the simple observation that these sorts of formulas continue to exist beyond linear algebra. Unfortunately, the new formulas are mostly intractable right now. In the ultra-classical world of cubic surfaces, however, Anand Deopurkar, Dennis Tseng and I did find some success, and my talk will focus primarily on this case. Our key formula unlocks things like: a general cubic surface arises 96120 times in a general 4-dimensional linear system of cubic surfaces, or a general cubic surface arises 42120 times as a hyperplane section of a general cubic threefold.