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.

Abstracts can be found below.

Spring 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.

Date Speaker Title
Thursday, Feb. 8 Matt Weaver (Notre Dame) Complete intersections and duality, with applications to Rees rings and tangent algebras
Thursday, Feb. 15 François Greer (Michigan State) Compactifying special cycles
Thursday, Mar. 7 Takumi Murayama (Purdue) TBA
Thursday, Mar. 14 No seminar (Spring break)
Thursday, Mar. 21 Ralph Kaufmann (Purdue) TBA
Thursday, Apr. 18 Fan Qin (Beijing Normal University) TBA

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.

Date Speaker Title
Thursday, Aug. 31 Alessandra Costantini (Oklahoma State) Rees algebras of linearly presented ideals
Wednesday, Sep. 6, 4-5pm
129 Hayes-Healy Hall
Department Colloquium
Eric Riedl (Notre Dame) Geometric Manin's and Lang's Conjectures
Thursday, Sep. 7 Keller VandeBogert (Notre Dame) The Total Rank Conjecture in Characteristic Two
Thursday, Sep. 14 Anand Patel (Oklahoma State) Counting cubic surfaces
Thursday, Oct. 5 Wenbo Niu (Arkansas) Fundamental forms of algebraic varieties
Friday, Oct. 6, 4-5pm
129 Hayes-Healy Hall
Department Colloquium
Claudia Polini (Notre Dame) Rees algebras
Thursday, Oct. 12 Ethan Reed (Notre Dame) An Isomorphism Theorem of Arithmetic Complexes
Thursday, Oct. 19 No seminar (Fall break)
Thursday, Oct. 26 Justin Campbell (Chicago) Langlands duality on the Beilinson-Drinfeld Grassmannian
Thursday, Nov. 2 Max Weinreich (Harvard) TBA
Thursday, Nov. 9 Karthik Ganapathy (Michigan) GL-equivariant modules over the infinite variable polynomial ring in positive characteristic
Thursday, Nov. 16
127 Hayes-Healy Hall, 2:30-3:30pm
Note special time / location
Tamanna Chatterjee (Notre Dame) Parabolic induction and parity sheaves for classical groups
Thursday, Nov. 23 No seminar (Thanksgiving)
Thursday, Nov. 30
127 Hayes-Healy Hall, 2:30-3:30pm
Note special time / location
Mahrud Sayrafi (Minnesota) Splitting of vector bundles on toric varieties
Thursday, Dec. 7
127 Hayes-Healy Hall, 2:30-3:30pm
Note special time / location
Teresa Yu (Michigan) Standard monomial theory modulo Frobenius in characteristic two

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.

Oct. 5, 2023

Speaker
Wenbo Niu (Arkansas)
Title
Fundamental forms of algebraic varieties
Abstract
Fundamental forms can be thought of as linear systems attached to the projectivization of Zariski tangent space at a nonsingular point of a variety. It was developed by the method of moving frames in differential geometry. In 1979, Griffiths-Harris used fundamental forms to study geometry of algebraic varieties and observed some vanishing phenomena. In this talk, I discuss a purely algebraic approach to the theory of fundamental forms without using moving frames. Furthermore, I will extend the vanishing of fundamental obtained by Griffiths-Harris and Landsberg to arbitrary order of fundamental forms. This is a joint work with L. Ein.

Oct. 12, 2023

Speaker
Ethan Reed (Notre Dame)
Title
An Isomorphism Theorem of Arithmetic Complexes
Abstract
We consider generalizations of certain arithmetic complexes appearing in work of Raicu and VandeBogert in connection with the study of stable sheaf cohomology on flag varieties. Defined over the ring of integer valued polynomials, we prove an isomorphism of these complexes as conjectured by Gao, Raicu, and VandeBogert. In particular, this gives a more conceptual proof of an identification between the stable sheaf cohomology of hook and two column partition Schur functors applied to the cotangent sheaf of projective space. This talk is based on joint work with Luca Fiorindo, Shahriyar Roshan-Zamir, and Hongmiao Yu.

Oct. 26, 2023

Speaker
Justin Campbell (Chicago)
Title
Langlands duality on the Beilinson-Drinfeld Grassmannian
Abstract
In recent joint work with Sam Raskin, we describe various categories of equivariant sheaves on the Beilinson-Drinfeld affine Grassmannian in Langlands dual terms. I will give an overview of this story, emphasizing some of the interesting infinite-dimensional geometry which occurs.

Nov. 9, 2023

Speaker
Karthik Ganapathy (Michigan)
Title
GL-equivariant modules over the infinite variable polynomial ring in positive characteristic
Abstract
In the presence of a large group action, even non-noetherian rings sometimes behave like noetherian rings. For example, Cohen proved that every symmetric ideal in the infinite variable polynomial ring is generated by the orbit of finitely many polynomials. In this talk, I will introduce a nice and tractable class of modules over the infinite variable polynomial ring motivated by Cohen's theorem and explain how their structure becomes more complicated in positive characteristic.

Nov. 16, 2023

Speaker
Tamanna Chatterjee (Notre Dame)
Title
Parabolic induction and parity sheaves for classical groups
Abstract
Parity sheaves are some constructible complexes defined on some stratified space where the strata satisfies some parity vanishing conditions. They are introduced by Carl Mautner, Daniel Juteau and Geordie Williamson in 2014. In characteristic 0 they coincide with the intersection cohomology complexes but in positive characteristic they are new and important objects. On flag variety they can be used as the "p-canonical basis" for Hecke algebras. It was noticed that on finite flag variety as well as affine grassmannian the parity sheaves correspond to the tilting sheaves. One expectation was to find similar relation on nilpotent cone, which Achar and Mautner started exploring in 2012. Another expectation was to understand the modular Springer correspondence in terms of parity sheaves. To study that one important conjecture has to be solved was made by Mautner. It says the parabolic induction functor defined on the nilpotent cones must preserves parity complexes. We break down this conjecture in two pieces and first try to prove that the parabolic induction functor sends parity sheaves associated to a cuspidal pair to a parity complex for classical groups. This is a ongoing project with Pramod N. Achar.

Nov. 30, 2023

Speaker
Mahrud Sayrafi (Minnesota)
Title
Splitting of vector bundles on toric varieties
Abstract
In 1964, Horrocks proved that a vector bundle on a projective space splits as a sum of line bundles if and only if it has no intermediate cohomology. Generalizations of this criterion, under additional hypotheses, have been proven for other toric varieties, for instance by Eisenbud-Erman-Schreyer for products of projective spaces, by Schreyer for Segre-Veronese varieties, and Ottaviani for Grassmannians and quadrics. This talk is about a splitting criterion for arbitrary smooth projective toric varieties.

Dec. 7, 2023

Speaker
Teresa Yu (Michigan)
Title
Standard monomial theory modulo Frobenius in characteristic two
Abstract
Over a field of characteristic zero, standard monomial theory and determinantal ideals provide an explicit decomposition of polynomial rings into simple GL_n-representations, which have characters given by Schur polynomials. In this talk, we present work towards developing an analogous theory for polynomial rings over a field of characteristic two modulo a Frobenius power of the maximal ideal generated by all variables. In particular, we obtain a filtration by modular GL_n-representations whose characters are given by certain truncated Schur polynomials, thus proving a conjecture by Gao-Raicu-VandeBogert in the characteristic two case. This is joint work with Laura Casabella.

Feb. 8, 2024

Speaker
Matt Weaver (Notre Dame)
Title
Complete intersections and duality, with applications to Rees rings and tangent algebras
Abstract
The Rees algebra plays a central role in both algebraic geometry and commutative algebra, as it allows one to encode the data on the powers of an ideal, along with their syzygies. In this talk however, we consider the more general notion of Rees algebras of modules, which tend to be much more enigmatic. Typically one attempts to approximate these rings by a more understood algebra, namely the symmetric algebra and hopes to measure how much these two rings differ. In this talk, we associate a module measuring this difference and we show it is dual to the symmetric algebra when this is a bigraded complete intersection ring. We then discuss applications to determining the equations defining Rees rings of modules over complete intersection rings and, in particular, modules of K\"ahler differentials and their associated tangent algebras.

Feb. 15, 2024

Speaker
François Greer (Michigan State)
Title
Compactifying special cycles
Abstract
A classical theorem of Borcherds and Zhang states that the cycle classes of Hodge loci in a moduli space of K3-type Hodge structures form the coefficients of a modular form. We investigate how well this theorem survives upon passing to a smooth compactification. As an application, we sketch a counterexample to the Severi Problem for rational surfaces.