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.

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.

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.

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.

Feb. 15, 2023

Speaker

Vaibhav Pandey (Purdue)

Title

Linkage and F-regularity of generic determinantal rings

Abstract

We prove that the generic link of a generic determinantal ring of maximal minors is strongly F-regular. In the process, we strengthen a result of Chardin and Ulrich. They showed that the generic residual intersections of a complete intersection ring with rational singularities again have rational singularities. We show that they are, in fact, strongly F-regular. In the mid 1990s, Hochster and Huneke showed that generic determinantal rings are strongly F-regular. However, their proof is quite involved. The techniques that we discuss will allow us to give a new and simple proof of the F-regularity of generic determinantal rings defined by maximal minors. Time permitting, we will also share a new proof of the F-regularity of determinantal rings defined by minors of any size. This is joint work with Yevgeniya Tarasova.

Mar. 1, 2023

Speaker

Maya Banks (Wisconsin)

Title

Boij-Söderberg Conjectures for Differential Modules

Abstract

Boij-Söderberg theory gives a combinatorial description of numerical invariants of modules over the polynomial ring and vector bundles on projective space. We posit that a similar combinatorial description can be given for analogous numerical invariants of graded differential modules, which are natural generalizations of chain complexes. In this talk, we will define graded differential modules over the polynomial ring and their numerical invariants, comparing and contrasting their behavior with that of free resolutions, and explore an analog of the Boij-Söderberg conjectures for this setting. We will discuss some partial results in support of these conjectures, including a duality between the Betti numbers of differential modules and the "absolute Hilbert function" of coherent sheaves on projective space.

Mar. 8, 2023

Speaker

Josh Pollitz (Utah)

Title

Frobenius push forwards and generators for the derived category

Abstract

By now it is quite classical that one can understand singularities in prime characteristic local algebra/algebraic geometry, through properties of the Frobenius endomorphism. A foundational result illustrating this is the celebrated theorem of Kunz characterizing the regularity of a noetherian scheme (in prime characteristic) in terms of whether a Frobenius push forward on that scheme is flat. In this talk, I'll discuss a structural explanation of, that also recovers, the theorem of Kunz and other theorems of this ilk. Namely, I’ll discuss recent joint work with Ballard, Iyengar, Lank, and Mukhopadhyay where we show that over an F-finite noetherian scheme of prime characteristic high enough Frobenius push forwards generate the bounded derived category.

Mar. 22, 2023

Speaker

Peter Koymans (Michigan)

Title

Two-step nilpotent extensions are not anabelian

Abstract

Neukirch proved that a number field is determined, up to isomorphism, by its absolute Galois group (as profinite group). This was later refined by Uchida who showed that one may instead take the Galois group of the maximal solvable extension. In this talk we are interested in the maximal nilpotent extension of class 2, and we will see that a number field is no longer determined by the corresponding Galois group. This is joint work with Carlo Pagano.

Mar. 29, 2023

Speaker

András Lőrincz (Oklahoma)

Title

On the collapsing of homogeneous bundles

Abstract

I present results on the geometry of equivariant, proper maps from homogeneous bundles over flag varieties, called collapsing maps. Kempf showed that is completely the image of a collapsing has rational singularities in characteristic zero when the bundle is completely reducible. We extend this to positive characteristics showing that such an image is strongly F-regular if its coordinate ring has a good filtration. We further prove that the restrictions of such collapsing maps to Schubert varieties are F-rational in positive characteristic and have rational singularities in characteristic zero. These results give a uniform, characteristic-free approach for the study of the geometry of some remarkable varieties, such as: multicones over Schubert varieties, various determinantal varieties in spaces of matrices, Buchsbaum-Eisenbud varieties of complexes, subspace varieties, higher rank varieties.

Apr. 5, 2023

Speaker

Eric Riedl (Notre Dame)

Title

Non-free curves and Geometric Manin's Conjecture

Abstract

One important way to study a variety is to study the curves on it. Free curves, those whose restricted tangent bundle has vanishing H^1, are particularly nice. They are smooth points of the space Hom(B,X) of morphisms from B to X, and they lie on a unique component of Hom(B,X) of the expected dimension. Nonfree curves, on the other hand, are more mysterious, and the components of Hom(B,X) consisting entirely of nonfree curves are particularly challenging to study. In this talk, we provide a geometric classification of the ways that a family of curves can be nonfree. In the more general setting of a Fano fibration, we show that these curves come from a-covers of X, and we show that the set of such a-covers up to equivalence form a bounded family. This proves the first of Batyrev's heuristics that make up Geometric Manin's Conjecture for Fano fibrations over arbitrary base curves and provides strong evidence for Lehmann, Sengupta and Tanimoto's proposed characterization of the thin set in Manin's Conjecture. This is joint work with Brian Lehmann and Sho Tanimoto.

Apr. 19, 2023

Speaker

Tymoteusz Chmiel (Jagiellonian University)

Title

Koszul modules of Kac-Moody Lie algebras and their resonance varieties

Abstract

I will start with a brief introduction to the theory of Koszul modules. Then I will describe the construction of Kac-Moody Lie algebras and define their associated Koszul modules. These modules can be nilpotent (which is equivalent to the vanishing of their resonance) or of infinite length. I will give an equivalent criterion for a Kac-Moody Koszul module to be nilpotent. Finally, I will mention how one can compute resonance varieties in the non-nilpotent cases.

Apr. 26, 2023

Speaker

Pallav Goyal (Chicago)

Title

Almost commuting variety and quantum Hamiltonian reduction

Abstract

I’ll be talking about the notions of commuting varieties and almost commuting varieties of Lie algebras in Types A and C. Then I’ll quantize these notions and discuss their applications to the representation theory of rational Cherednik algebras via Hamiltonian reduction.

May. 3, 2023

Speaker

Thomas Brazelton (Pennsylvania)

Title

Equivariant enumerative geometry

Abstract

Classical enumerative geometry asks geometric questions of the form "how many?" and expects an integral answer. For example, how many circles can we draw tangent to a given three? How many lines lie on a cubic surface? The fact that these answers are well-defined integers, independent upon the initial parameters of the problem, is Schubert’s principle of conservation of number. In this talk we will outline a program of "equivariant enumerative geometry", which wields equivariant homotopy theory to explore enumerative questions in the presence of symmetry. Our main result is equivariant conservation of number, which states roughly that the orbits of solutions to an equivariant enumerative problem are conserved. We leverage this to compute the S4 orbits of the 27 lines on any smooth symmetric cubic surface.