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.

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.

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.

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.

Mar. 7, 2024

Speaker

Takumi Murayama (Purdue)

Title

Boutot’s theorem and vanishing theorems for Zariski–Riemann spaces

Abstract

Let S be a regular ring. By work of Hochster–Roberts, Boutot, Smith, Hochster–Huneke, Schoutens, and Heitmann–Ma, every pure subring of S is pseudo-rational, and in particular, Cohen–Macaulay. This applies for example to rings of invariants of linearly reductive groups. For pure maps R → S of rings of finite type over the complex numbers, Boutot's result is even stronger: If S has rational singularities, then R has rational singularities. In this talk, I will discuss my generalization of Boutot’s theorem which applies to arbitrary Noetherian Q-algebras R and S. The key new ingredient is a new vanishing theorem for Zariski–Riemann spaces of Noetherian schemes in equal characteristic zero. My vanishing theorem has many applications. For example, it implies relative vanishing theorems and the existence (proved jointly with Shiji Lyu) of the relative minimal model program with scaling for algebraic spaces, formal schemes, and both complex and non-Archimedean analytic spaces.

Mar. 14, 2024

Speaker

Ralph Kaufmann (Purdue)

Title

The algebra of categories and its application

Abstract

There is a way to encode categories as bimodule algebras which is particularly suited to treat equivariant aspects. Utilizing this point of view many operations known from algebra become readily available for categories, such as coalgebras, free (co) modules, bar/cobar resolutions and so on. Moving to monoidal categories adds another multiplicative structure and directly leads into the theory of PROPs, Feynman categories and unique factorization categories. Feynman categories are then easily defined as coming from those bimodules whose free modules are monoidal. Bi- or Hopf algebras and Connes-Kreimer type B_+ operators as CoHochschild cocylces also naturally appear in this context. In a second direction one can consider the generalization of quadratic algebras and Koszul duality. In the newest developments there is a conjecture relation between cubical -the right notion of quadratic- structures and cluster-like transformations, which we will address if time permits.

Apr. 16, 2024

Speaker

Fan Qin (Beijing Normal University)

Title

Analogs of dual canonical bases for cluster algebras from Lie theory

Abstract

The (quantized) coordinate rings of many interesting varieties from Lie theory are (quantum) cluster algebras. We construct the common triangular bases for these algebras. Such bases provide analogs of the dual canonical bases, whose existence has been long expected in cluster theory. For symmetric Cartan matrices, they are positive and admit monoidal categorification after base change.

Apr. 25, 2024

Speaker

Dmitri Voloshyn (Korea)

Title

Generalized cluster structures on the special linear group

Abstract

The Gekhtman-Shapiro-Vainshtein conjecture (the GSV conjecture) states that for any given simple complex algebraic group G and any Poisson bracket from the Belavin-Drinfeld class, there exists a compatible generalized cluster structure. In this talk, I will review the process of constructing compatible generalized cluster structures, as well as the current state-of-the-art on the GSV conjecture. After that, I will describe a construction of generalized cluster structures on \mathrm{SL}_n compatible with Poisson brackets induced from the Poisson dual of \mathrm{SL}_n endowed with the Poisson structure determined by a BD triple of type A_{n-1}. I will also describe the associated family of birational quasi-isomorphisms. The talk will be based on the preprint arXiv:2312.04859 (joint work with M. Gekhtman).