Algebraic Geometry/Commutative Algebra Seminar, 2024–2025

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

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.

Jan. 24, 2025

Speaker

Adam Boocher (San Diego)

Title

From Classical Commutative Algebra to Some Diophantine Equations

Abstract

In a first course in commutative algebra one might encounter the "Principal Ideal Theorem" or the "Auslander-Buchsbaum Formula". It turns out that these are both implied by a longstanding conjecture about lower bounds for ranks of syzygies - the Buchsbaum-Eisenbud-Horrocks Rank Conjecture. In this talk I'll discuss historical progress on the conjecture as well as a related (and rather mysterious) conjecture about even larger bounds. Later, I'll discuss some recent work about what happens if one looks at the special case of pure modules, using techniques from Boij-Soederberg Theory. This approach leads to some interesting diophantine equations, which may shed light on the original conjectures. This is joint work with my two undergraduate students Noah Huang and Harrison Wolf.

Feb. 6, 2025

Speaker

Eric Riedl (Notre Dame)

Title

Free curves in singular varieties

Abstract

Rational curves play a critical role in understanding the birational geometry of varieties. Free curves are the easiest to work with, but on Fano varieties that are even mildly singular, it remains an open question whether these free rational curves exist. In this talk, we discuss free curves of higher genus. Using some ideas on stability of vector bundles, we show that any klt Fano variety has these higher-genus free curves. We then use the existence of these free curves to get some applications, including the existence of free rational curves in terminal Fano threefolds, the lengths of extremal rays of the cone of curves, and studying the fundamental group of the smooth locus of a terminal variety. This is joint work with Eric Jovinelly and Brian Lehmann.

Feb. 20, 2025

Speaker

Sam Evens (Notre Dame)

Title

Compactification of a Cartan subalgebra and additive toric varieties

Abstract

The wonderful compactification of a semisimple Lie group G of adjoint type is a well-studied projective variety. If H is a maximal torus of G, then the closure of H in the wonderful compactification is a smooth toric variety with fan given by the Weyl chamber decomposition. We discuss a degeneration of the wonderful compactification that arises in Poisson geometry, where the roles of G and H are played by their Lie algebras $\fg$ and $\fh.$ While the geometry of the compactification of $\fg$ is not well-understood, we show that the closure of $\fh$ can be understood using ideas from the theory of matroid Schubert varieties, and is normal and Cohen-Macaulay, and has a paving by affines, which are given by orbits of $\fh$ regarded as an algebraic group. If time permits, we will discuss the interplay between our results and Stirling numbers and the Dowling lattice. This talk is based on joint work with Yu Li.

Mar. 17, 2025

Speaker

Emanuela Marangone (Manitoba)

Title

Cohomology of line bundles on the incidence correspondence

Abstract

The study of the cohomology of line bundles on (partial) flag varieties is an important problem at the intersection of algebraic geometry, commutative algebra, and representation theory. Over fields of characteristic zero, this is well-understood thanks to the Borel--Weil--Bott theorem, but in positive characteristics, it remains largely open. In this talk, we will focus on the incidence correspondence, the partial flag variety parameterizing pairs consisting of a point in projective space and a hyperplane containing it. I will describe joint work with Claudiu Raicu, Annet Kyomuhangi, and Ethan Reed, where we derive a recursive formula for the characters of the cohomology of line bundles on the incidence correspondence in positive characteristic. In characteristic 2, we provide a closed-form formula. I will also explain how this cohomology calculation characterizes the Weak Lefschetz Property (WLP) for Artinian monomial complete intersections and present a test for the WLP in characteristic 2.

Mar. 20, 2025

Speaker

Philip Engel (UIC)

Title

Boundedness theorems for abelian fibrations

Abstract

I will report on work-in-progress, with Filipazzi, Greer, Mauri, and Svaldi on boundedness results for abelian fibrations. We will discuss a proof that irreducible Calabi-Yau varieties admitting an abelian fibration are birationally bounded in a fixed dimension; and similarly, that Lagrangian fibrations of symplectic varieties, in a fixed dimension, are analytically bounded.

Apr. 3, 2025

Speaker

Mike Perlman (Minnesota)

Title

Effective vanishing for toric vector bundles

Abstract

A toric vector bundle on a toric variety is a vector bundle endowed with a torus action compatible with the one on the underlying space. Almost all familiar examples of bundles on toric varieties are toric, from (co)tangent bundles to kernel bundles associated to complete linear series, which control syzygies of the corresponding embedding. In contrast to the case of line bundles, these are not purely combinatorial objects, as they also depend on linear algebraic information associated to cones known as the Klyachko data. We will discuss a new effective vanishing result for twists of toric vector bundles by line bundles, taking into account both the linear algebra and combinatorics via Batyrev’s primitive relations. This vanishing recovers many familiar results, and our techniques lead to new ways to study N_{p} properties for smooth projective toric varieties. This is joint work in progress with Gregory G. Smith.