Algebraic Geometry/Commutative Algebra Seminar, 2024–2025

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

Spring Schedule

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

Fall Schedule

The seminar will meet on Fridays, 3–4pm in 258 Hurley, unless otherwise noted. Related events are also listed below.

Abstracts

Sep. 6, 2024

Speaker

Xianglong Ni (Notre Dame)

Title

Herzog classes of grade three licci ideals

Abstract

By work of Buchweitz and Herzog, there is a well-defined classification of licci ideals up to deformation. The equivalence classes obtained in this manner are called Herzog classes. For grade 2 perfect ideals (all of which are licci) the Herzog class of I is determined by its minimal number of generators, i.e. the vector space dimension of I/mI. Furthermore, the class of a linked ideal K:I can be inferred from the dimension of the subspace (K+mI)/mI. Assuming equicharacteristic zero, we generalize this to grade 3 licci ideals, where we can describe all Herzog classes with the assistance of representation theory. In this setting, the class of K:I depends on the incidence of (K+mI)/mI with a distinguished partial flag on I/mI. This is based on ongoing joint work with Lorenzo Guerrieri and Jerzy Weyman.

Sep. 20, 2024

Speaker

Keller VandeBogert (Notre Dame)

Title

From Total Positivity to Pure Free Resolutions

Abstract

Pólya frequency sequences are ubiquitous objects with a surprising number of connections to many different areas of mathematics. It has long been known that such sequences admit a “duality" operator that mimics the duality of a Koszul algebra and its quadratic dual, but the precise connection between these notions turns out to be quite subtle. In this talk, we will see how the equivariant analogue of Pólya frequency is closely related to the problem of constructing Schur functors “with respect to" an algebra. We will moreover see how these ideas come together to understand the problem of extending Boij--Soederberg theory to other classes of rings, with particular attention given to the case of quadric hypersurfaces. This is based on joint work with Steven V. Sam.

Oct. 4, 2024

Speaker

Claudia Polini (Notre Dame)

Title

Infinite free resolutions: a finite story for Golod rings

Abstract

We prove a surprising finiteness result for Golod rings: All the syzygies of the residue field of a Golod ring are direct sums of a finite set of indecomposable modules. This finetistic behavior is diametrically opposite to the behavior of Artinian Gorenstein rings, where each syzygy of the residue field is indecomposable. In addition we describe in detail the case of any local ring of embedding dimension two that is not a complete intersection. In this case, all syzygy modules are direct sums of only three possible modules, the residue field, the maximal ideal, and the dual of the maximal ideal.

Oct. 11, 2024

Speaker

Dave Swinarski (Fordham)

Title

Singular curves in Mukai's model of $\bar{M}_7$

Abstract

In 1995 Mukai showed that a general smooth genus 7 curve can be realized as the intersection of the orthogonal Grassmannian OG(5,10) in P^15 with a six-dimensional projective linear subspace, and that the GIT quotient Gr(7,16)//Spin(10) is a birational model of the moduli space of curves $\bar{M}_7$. Which singular objects appear on the boundary of Mukai's model? As a first step in this study, calculations in Macaulay2 and Magma are used to find and analyze linear spaces yielding three singular curves: a 7-cuspidal curve, the balanced ribbon of genus 7, and a family of genus 7 graph curves.

Nov. 1, 2024

Speaker

Shuddhodan Vasudevan (Notre Dame)

Title

Revisiting the basic lemma

Abstract

The basic lemma due to Beilinson is a fundamental result of the topology of algebraic varieties. In this talk, I will introduce the Brylinski-Radon transformation and discuss its properties and applications to the basic lemma and its corollaries. Time permitting, we will also discuss some arithmetic applications of our results. This is joint work with Ankit Rai.

Nov. 8, 2024

Speaker

Juan Migliore (Notre Dame)

Title

Intersections of curves in P^4

Abstract

I'll speak on joint work (almost finished) with Luca Chiantini (Siena), \L ucja Farnik (Krakow), Giuseppe Favacchio (Palermo), Brain Harbourne (Nebraska), Tomasz Szemberg (Krakow) and Justyna Szpond (Krakow). In how many points can two irreducible, non-degenerate curves, $C_1$ and $C_2$, of degrees $d_1$ and $d_2$ respectively, meet in projective space? For $\mathbb P^3$ we have a pretty good picture, thanks to work of Diaz (1986) and of Giuffrida (1986). Refinements were made by Hartshorne--Mir\'o-Roig (2015) requiring the curves to be arithmetically Cohen-Macaulay, and by Chiantini-Migliore (2021) allowing the curves to be reducible. According to Hartshorne and Mir\'o-Roig, ``There seems to be scant attention to these questions in the literature.'' The Diaz-Giuffrida bound is $(d_1-1)(d_2-1) +1$ and occurs exactly when one curve is of type $(d_1-1,1)$ on a smooth quadric surface $Q$ (a surface of minimal degree), and $C_2$ is of type $(1,d_2-1)$ on the same $Q$. If the curves do not both lie on a smooth quadric, the curves cannot achieve the bound (there are open questions about this). In the current project we ask the analogous question for curves in $\mathbb P^4$, and we get some surprising similarities and differences from the situation in $\mathbb P^3$, which will be described in this talk. After a computation for intersections of curves on a cubic surface (a surface of minimal degree), our approach turns to a study of the arithmetic genus of $C_1 \cup C_2$ and a very careful study of the Hilbert function of the general hyperplane section, $\Gamma$, of $C_1 \cup C_2$. Along the way we need to ``lift" properties of $\Gamma$ to $C_1 \cup C_2$, which we do using a theorem of Huneke-Ulrich and Strano.

Nov. 15, 2024

Speaker

Jacob Zoromski (Notre Dame)

Title

Monomial Cycles in Koszul Homology

Abstract

Resolutions over a quotient R of a polynomial ring are closely related to an algebra structure on Tor(R,k). When R is a quotient of a monomial ideal, "monomial cycles" in Tor(R,k) are abundant. I show their vanishing is governed by the combinatorics of full simplicial matroids, and as a result classify Golod monomial ideals in four variables.

Nov. 22, 2024

Speaker

Alexandru Chirvăsitu (Buffalo)

Title

Secant slices as symplectic leaves of moduli-space Poisson structures

Abstract

The non-commutative algebras $Q_{n,k}(E,\eta)$, introduced by Feigin and Odesskii in the course of generalizing Sklyanin's work, depend on two coprime integers $n>k\ge 1$, an elliptic curve $E$ and a point $\eta\in E$. The degeneration $\eta\to 0$ collapses $Q_{n,1}(E,\eta)$ to the polynomial ring in $n$ variables, and one obtains in this fashion a homogeneous Poisson bracket on that polynomial ring and hence a Poisson structure on the projective space $\mathbb{P}^{n-1}$. The symplectic leaves attached to that structure have received some attention in the literature, including from Feigin and Odesskii themselves and, more recently, Hua and Polishchuk. The talk revolves around various results on these symplectic leaves: their concrete description as moduli spaces of sheaf extensions on the elliptic curve $E$, the attendant realization as GIT quotients, resulting good properties (like smoothness) which follow from this without appealing to the symplectic machinery, etc. (joint with Ryo Kanda and S. Paul Smith)

Dec. 6, 2024

Speaker

Grant Barkley (Harvard)

Title

The combinatorial invariance conjecture

Abstract

Let u and v be two permutations of the numbers 1,...,n. Associated to u and v is a polynomial P_uv, called the Kazhdan-Lusztig polynomial, which encodes numerical invariants that are central in geometric representation theory. The coefficients of P_uv simultaneously describe the singularities of Schubert varieties, the structure of Hecke algebras, and the representation theory of Lie algebras. Associated to u and v is another object, the Bruhat graph of (u,v), which is a directed graph describing the transpositions taking u to v. The combinatorial invariance conjecture (CIC) of Dyer and Lusztig asserts that the Bruhat graph of (u,v) uniquely determines P_uv. Recently, Geordie Williamson and Google DeepMind applied machine learning techniques to this problem. Using those techniques, they conjectured an explicit recursion that would compute P_uv from the Bruhat graph and thereby prove the CIC. In joint work with Christian Gaetz, we prove the Williamson-DeepMind conjecture in the case where u is the identity permutation. Along the way, we prove two new identities for the Kazhdan-Lusztig R polynomials, one of which implies new cases of the CIC.