# Algebraic Geometry/Commutative Algebra Seminar, 2022–2023

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 Wednesdays, 3–4pm, either on Zoom or in 258 Hurley, unless otherwise noted. Related events are also listed below.

Date Speaker Title
Wednesday, Jan. 25 Anna Bot (Basel) A smooth complex rational affine surface with uncountably many nonisomorphic real forms
Wednesday, Feb. 1 Evan O'Dorney (Notre Dame) Diophantine Approximation on Conics
Wednesday, Feb. 8 Uwe Nagel (Kentucky) Schemes arising from hypersurface arrangements
Wednesday, Feb. 15 Vaibhav Pandey (Purdue) TBA
Wednesday, Feb. 22
Wednesday, Mar. 1 Maya Banks (Wisconsin) TBA
Wednesday, Mar. 8
Wednesday, Mar. 15 No seminar (Spring break)
Wednesday, Mar. 22
Wednesday, Mar. 29
Wednesday, Apr. 5
Wednesday, Apr. 12
Wednesday, Apr. 19 Tymoteusz Chmiel (Jagiellonian University) TBA
Wednesday, Apr. 26
Wednesday, May. 3

## Fall Schedule

The seminar will meet on Tuesdays, 2:30–3:30pm, either on Zoom or in 258 Hurley, unless otherwise noted. Related events are also listed below.

Date Speaker Title
Tuesday, Sep. 6 Steven Karp (Notre Dame) Wronskians, total positivity, and real Schubert calculus
Tuesday, Sep. 13 Joe Waldron (Michigan State) Purely inseparable Galois theory
Tuesday, Sep. 20 Mark Skandera (Lehigh University) Type-BC character evaluations and P-tableaux
Tuesday, Sep. 27 Joe Cummings (Notre Dame) A Fano compactification of the free group character variety
Tuesday, Oct. 4 Joshua Mundinger (Chicago) Quantization of restricted Lagrangian subvarieties in positive characteristic
Tuesday, Oct. 11 Bernd Ulrich (Purdue) Duality and blowup algebras
Tuesday, Oct. 11
5-6pm
Hunter Simper (Purdue) Local Cohomology Modules of Thickenings of Ideals of Maximal Minors
Tuesday, Oct. 18 No seminar (Fall break)
Tuesday, Oct. 25 Cheng Meng (Purdue) Multiplicities in flat local extensions
Tuesday, Nov. 1 Jenny Kenkel (Michigan) Lengths Of Local Cohomology Using Some Surprising Hilbert Kunz Functions
Tuesday, Nov. 8 Brandon Alberts (Eastern Michigan) A Random Group with Local Data
Tuesday, Nov. 15 Nitin Chidambaram (Edinburgh) r-th roots – better negative than positive
Tuesday, Nov. 22 Mihai Fulger (Connecticut) Tangent cones of pluritheta divisors on abelian threefolds
Tuesday, Nov. 29 Martha Precup (Washington University in St. Louis) The cohomology of nilpotent Hessenberg varieties and the dot action representation
Tuesday, Dec. 6 Lena Ji (Michigan) Finite order birational automorphisms of Fano hypersurfaces

## Abstracts

### Sep. 6, 2022

Speaker
Steven Karp (Notre Dame)
Title
Wronskians, total positivity, and real Schubert calculus
Abstract
The totally positive flag variety is the subset of the complete flag variety Fl(n) where all Plücker coordinates are positive. By viewing a complete flag as a sequence of subspaces of polynomials of degree at most n-1, we can associate a sequence of Wronskian polynomials to it. I will present a new characterization of the totally positive flag variety in terms of Wronskians, and explain how it sheds light on conjectures in the real Schubert calculus of Grassmannians. In particular, a conjecture of Eremenko (2015) is equivalent to the following conjecture: if V is a finite-dimensional subspace of polynomials such that all complex zeros of the Wronskian of V are real and negative, then all Plücker coordinates of V are positive. This conjecture is a totally positive strengthening of a result of Mukhin, Tarasov, and Varchenko (2009), and can be reformulated as saying that all complex solutions to a certain family of Schubert problems in the Grassmannian are real and totally positive.

### Sep. 13, 2022

Speaker
Joe Waldron (Michigan State)
Title
Purely inseparable Galois theory
Abstract
Given a field K of characteristic p, a classical result of Jacobson provides a Galois correspondence between finite purely inseparable subfields of K of exponent one (i.e. those which contain K^p), and sub-restricted Lie algebras of Der(K). I will discuss joint work with Lukas Brantner in which we extend this Galois correspondence to subfields of arbitrary exponent using methods from derived algebraic geometry.

### Sep. 20, 2022

Speaker
Mark Skandera (Lehigh University)
Title
Type-BC character evaluations and P-tableaux
Abstract
For w in S_n corresponding to a smooth type-A Schubert variety X(w), there exists a poset P = P(w) which provides a combinatorial interpretation of the Kazhdan-Lusztig basis element C_w(1) in Z[S_n]. Evaluations of characters at C_w(1) may be computed by counting special P-tableaux. We state a type-BC analog of these results. Namely, for w in the hyperoctahedral group B_n such that the type-B and type-C Schubert varieties are simultaneously smooth, there exists a type-BC poset P = P(w) which provides a combinatorial interpretation of the Kazhdan-Lusztig basis element C_w(1) in Z[B_n]. Again, evaluations of characters at the Kazhdan-Lusztig basis element C_w(1) of Z[B_n] may be computed by counting special P-tableaux.

### Sep. 27, 2022

Speaker
Joe Cummings (Notre Dame)
Title
A Fano compactification of the free group character variety
Abstract
We show that a certain compactification of the free group character variety $\mathcal{X}(F_g, Sl_2(\mathbb{C}))$ is Fano. This compactification has been studied previously by Manon, and separately by Biswas, Lawton, and Ramras. Part of the proof involves the construction of a large family of integral reflexive polytopes arising from trivalent graphs of genus $g$. This project is joint with Chris Manon.

### Oct. 4, 2022

Speaker
Joshua Mundinger (Chicago)
Title
Quantization of restricted Lagrangian subvarieties in positive characteristic
Abstract
In order to answer basic questions in modular and geometric representation theory, Bezrukavnikov and Kaledin introduced quantizations of symplectic varieties in positive characteristic. These are certain noncommutative algebras over a field of positive characteristic which have a large center. I will discuss recent work describing how to construct an important class of modules over such algebras.

### Oct. 11, 2022

Speaker
Bernd Ulrich (Purdue)
Title
Duality and blowup algebras
Abstract
This talk is concerned with a classical problem in elimination theory, the determination of the implicit equations defining the graphs and images of rational maps between projective varieties. The problem amounts to identifying the torsion in the symmetric algebra of an ideal, and one technique to achieve this is based on a duality statement due to Jouanolou that expresses the torsion of a graded algebra in terms of a graded dual of this algebra. Unfortunately, Jouanlou duality requires the algebra to be Gorenstein, a rather restrictive hypothesis for symmetric algebras. In this talk, I introduce a generalized notion of Gorensteinness, which we call weakly Gorenstein, and prove that Jouanolou duality generalizes to this larger class of algebras. Surprisingly, the weak Gorenstein property is rather common and is satisfied, for instance, by symmetric algebras assuming, mainly, that these are Cohen-Macaulay. This leads to the solution of the implicitization problem for new classes of rational maps. The talk is based on joint work with Yairon Cid-Ruiz and Claudia Polini.

### Oct. 11, 2022

Speaker
Hunter Simper (Purdue)
Title
Local Cohomology Modules of Thickenings of Ideals of Maximal Minors
Abstract
Let $R$ be the ring of polynomial functions in $mn$ variables with coefficients in $\mathbb{C}$, where $m>n$. Set $X$ to be the matrix in these variables and $I$ the ideal of maximal minors of this matrix. I will discuss the R-module structure of $Ext^i_R(R/I^t,R)$ and $H_\frak{m}^{mn-i}(R)$.

### Oct. 25, 2022

Speaker
Cheng Meng (Purdue)
Title
Multiplicities in flat local extensions
Abstract
We introduce the notion of strongly Lech-independent ideals as a generalization of Lech-independent ideals defined by Lech and Hanes, and use this notion to derive inequalities on multiplicities of ideals. In particular we prove that if (R,m) and (S,n) are Noetherian local rings of the same dimension, S is a flat local extension of R,and up to completion S is standard graded over a field and I=mS is homogeneous, then e(R) is no greater than e(S).

### Nov. 1, 2022

Speaker
Jenny Kenkel (Michigan)
Title
Lengths Of Local Cohomology Using Some Surprising Hilbert Kunz Functions
Abstract
We investigate the lengths of certain local cohomology modules over polynomial rings. By fixing the degree component, and using the fact that the length of an Artinian ring is the same as that of its injective hull, we transform this into a question about rings of the form k[x_1, \dots, x_n]/(x_1^{k_k}, \dots, x_n^{k_n}) and the annihilator of x_1 + \cdots + x_n therein. We in particular use refinements of functions introduced by Han and Monsky. This was motivated by questions about behavior of the length of local cohomology with support in the maximal ideal of thickenings, that is, R/I^t as t grows.

### Nov. 8, 2022

Speaker
Brandon Alberts (Eastern Michigan)
Title
A Random Group with Local Data
Abstract
The Cohen--Lenstra heuristics describe the distribution of $\ell$-torsion in class groups of quadratic fields as corresponding to the distribution of certain random p-adic matrices. These ideas have been extended to using random groups to predict the distributions of more general unramified extensions in families of number fields (see work by Boston--Bush--Hajir, Liu--Wood, Liu--Wood--Zureick-Brown). Via the Galois correspondence, the distribution of unramified extensions is a specific example of counting number fields with prescribed ramification and bounded discriminant. As of yet, no constructions of random groups have been given in the literature to predict the answers to famous number field counting conjectures such as Malle's conjecture. We construct a "random group with local data" bridging this gap, and use it to describe new heuristic justifications for number field counting questions.

### Nov. 15, 2022

Speaker
Nitin Chidambaram (Edinburgh)
Title
r-th roots – better negative than positive
Abstract
I will talk about the construction and properties of a cohomological field theory (without a flat unit) that parallels the famous Witten r-spin class. In particular, one can view it as the negative r analogue of the Witten r-spin class. For r=2, it was constructed by Norbury in 2017 and called the Theta class, and we generalize this construction to any r. By studying certain deformations of this class, we prove relations in the tautological ring, and in the special case of r=2 they reduce to relations involving only Kappa classes (which were recently conjectured by Norbury-Kazarian).

In the second part of this talk, we will exploit the relation between cohomological field theories and the Eynard-Orantin topological recursion to prove W-algebra constraints satisfied by the descendant potential of the class. Furthermore, we conjecture that this descendant potential is the r-BGW tau function of the r-KdV hierarchy, and prove it for r=2 (thus proving a conjecture of Norbury) and r=3.

This is based on joint work with Elba Garcia-Failde and Alessandro Giacchetto.

### Nov. 22, 2022

Speaker
Mihai Fulger (Connecticut)
Title
Tangent cones of pluritheta divisors on abelian threefolds
Abstract
The Riemann singularity theorem computes the multiplicity of points on theta divisors on Jacobians in terms of dimensions of linear series on the curve. It is also interesting to study singularities of pluritheta divisors. On Jacobians of genus 3 curves C, the multiplicity at the origin of the difference divisor C-C determines whether the curve is hyperelliptic or not. We compute the infinitesimal Newton-Okounkov body of the principal polarization on some abelian threefolds. In particular this recovers asymptotic information about the tangent cones at the origin of pluritheta divisors. This is joint work with Victor Lozovanu.

### Nov. 29, 2022

Speaker
Martha Precup (Washington University in St. Louis)
Title
The cohomology of nilpotent Hessenberg varieties and the dot action representation
Abstract
In 2015, Brosnan and Chow, and independently Guay-Paquet, proved the Shareshian--Wachs conjecture, which links the combinatorics of chromatic symmetric functions to the geometry of Hessenberg varieties via a permutation group action on the cohomology ring of regular semisimple Hessenberg varieties. This talk will give a brief overview of that story and discuss how the dot action can be computed in all Lie types using the Betti numbers of certain nilpotent Hessenberg varieties. As an application, we obtain new geometric insight into certain linear relations satisfied by chromatic symmetric functions, known as the modular law. This is joint work with Eric Sommers.

### Dec. 6, 2022

Speaker
Lena Ji (Michigan)
Title
Finite order birational automorphisms of Fano hypersurfaces
Abstract
The birational automorphism group is a natural birational invariant associated to an algebraic variety. In this talk, we study the specialization homomorphism for the birational automorphism group. As an application, building on work of Kollár and work of Chen–Stapleton, we show that a very general n-dimensional complex hypersurface X of degree ≥ 5⌈(n+3)/6⌉ has no finite order birational automorphisms. This work is joint with N. Chen and D. Stapleton.

### Jan. 25, 2023

Speaker
Anna Bot (Basel)
Title
A smooth complex rational affine surface with uncountably many nonisomorphic real forms
Abstract
A real form of a complex algebraic variety X is a real algebraic variety whose complexification is isomorphic to X. Many families of complex varieties have a finite number of nonisomorphic real forms, but up until recently no example with infinitely many had been found. In 2018, Lesieutre constructed a projective variety of dimension six with infinitely many nonisomorphic real forms, and this year, Dinh, Oguiso and Yu described projective rational surfaces with infinitely many as well. In this talk, I’ll present the first example of a rational affine surface having uncountably many nonisomorphic real forms.

### Feb. 1, 2023

Speaker
Evan O'Dorney (Notre Dame)
Title
Diophantine Approximation on Conics
Abstract
Given a conic $\mathcal{C}$ over $\mathbb{Q}$, it is natural to ask what real points on $\mathcal{C}$ are most difficult to approximate by rational points of low height. For the analogous problem on the real line (for which the least approximable number is the golden ratio, by Hurwitz's theorem), the approximabilities comprise the classically studied Lagrange and Markoff spectra, but work by Cha--Kim and Cha--Chapman--Gelb--Weiss shows that the spectra of conics can vary. We provide notions of approximability, Lagrange spectrum, and Markoff spectrum valid for a general $\mathcal{C}$ and prove that their behavior is exhausted by the special family of conics $\mathcal{C}_n : XZ = nY^2$, which has symmetry by the modular group $\Gamma_0(n)$ and whose Markoff spectrum was studied in a different guise by A. Schmidt and Vulakh. The proof proceeds by using the Gross-Lucianovic bijection to relate a conic to a quaternionic subring of $\operatorname{Mat}^{2\times 2}(\mathbb{Z})$ and classifying invariant lattices in its $2$-dimensional representation.

### Feb. 8, 2023

Speaker
Uwe Nagel (Kentucky)
Title
Schemes arising from hypersurface arrangements
Abstract
Let $A$ be a hypersurface arrangement. We consider the Cohen-Macaulayness of two unmixed ideals that are related to the Jacobian ideal $J$ of $A$: the intersection of height two primary components of $J$ and the radical of $J$. In joint work with Migliore and Schenck we showed that in the case of a hyperplane arrangement these ideals are Cohen-Macaulay under a mild hypothesis. We discuss extensions for hypersurface arrangements obtained in joint work with Juan Migliore.