Algebraic Geometry/Commutative Algebra Seminar, 2025–2026

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 Thursdays, 3:30–4:30pm in 258 Hurley, unless otherwise noted. Related events are also listed below.

Abstracts

Sep. 4, 2025

Speaker

Debaditya Raychaudhury (Arizona)

Title

Singularities of Secant varieties

Abstract

Secant varieties are classical objects in algebraic geometry. Given a smooth projective variety inside a projective space, its secant variety is by definition the closure of the union of secant lines. It is almost always singular and sits inside the same projective space by its construction. In this talk, we will discuss the singularities of secant varieties when the embedding is sufficiently positive. In particular, we will study the Du Bois complex of secant varieties and will also discuss about its local cohomology modules. The results are obtained in various collaborations with Q. Chen, B. Dirks, S. Olano and L. Song.

Sep. 25, 2025

Speaker

Eric Jovinelly (Brown)

Title

Free Curves in Singular Varieties

Abstract

Rational curves are intricately linked to the birational geometry of varieties containing them. Certain curves, called free curves, have the nicest deformation properties. However, it is unknown whether mildly singular Fano varieties contain free rational curves in their smooth locus. In this talk, we discuss free curves of higher genus. Using recent results about tangent bundles, we prove that any klt Fano variety has higher genus free curves. We then use the existence of such free curves to get some applications: we prove the existence of free rational curves in terminal Fano threefolds; obtain an optimal upper bound on the length of extremal rays in the Kleiman-Mori cone of any klt pair; and study the fundamental group of the smooth locus of a Fano variety. This is joint work with Brian Lehmann and Eric Riedl.

Oct. 1, 2025

Speaker

Vasudevan Srinivas (Buffalo)

Title

What are the Bloch-Beilinson Conjectures?

Abstract

The Bloch-Beilinson Conjectures are among the deepest open questions today, relating aspects of algebraic geometry, algebraic K-theory and number theory. The conjectures have roots, on the one hand, in classical results (Euler, Riemann, Dedekind, Hilbert, Artin, etc.) on special values and zeroes of zeta functions, in the period up to the early 20th century. Another source, somewhat more recent (going on to the mid 1970's) is work of Tate, Iwasawa, Lichtenbaum, Quillen and Borel, which brought in the role of algebraic K-theory. A more recent inspiration, beginning with several key calculations of Bloch, relate these to algebraic geometry. Bloch's vision was articulated in a general, more precise form by Beilinson, around 1982, resulting in what are now called the Bloch-Beilinson Conjectures. There are also refinements (e.g. the Bloch-Kato conjectures). In fact there is tantalizing, but rather meagre, evidence to support these conjectures, in spite of over 40 years of effort by interested mathematicians. My lecture will attempt to give an accessible introduction to this important circle of ideas.

Oct. 2, 2025

Speaker

Vijaylaxmi Trivedi (Buffalo)

Title

Numerical characterizations for integral dependence of graded ideals

Abstract

Two nonzero ideals I ⊆ J ⊂ OX, where X denotes an affine normal variety, are intergrally dependent (or have the same integral closure) if IOX+ = J OX+, where X+ −→ X is the normalization of the blow up along J. This notion is used in singularity theory. In dealing with the questions which depend only on the integral closure of an ideal, it is often convenient to replace J by a smaller ideal I which has the same integral closure. The notion of integral closure makes sense in any Noetherian commutative ring. The integral dependence of ideals of finite colength in a local ring has a well known numerical characterization, namely equality of Hilbert-Samuel multiplicities, due to D. Rees. Attempts to give a numerical characterization for ideals which might not necessarily be of finite colength led to numerical invariants like j-multiplicity, ε-multiplicity which require computing the invariant at several localizations, hence not readily amenable to computations. Also there exists a notion of multiplicity sequence which gives a numerical characterization of integral dependence. In this talk we give a list of numerical characterizations of the integral dependence of I and J in graded setup. In particular any of the multiplicities, namely, the polar multiplicities, the ε-multiplicities or the j-multiplicities of the truncated ideals I[Y]≥c and J[Y]≥c in R[Y] characterizes the integral dependence of I and J. A novelty of this approach is that it does not involve localization and only requires checking computable and well-studied invariants like Hilbert-Samuel multiplicities. Apart from several well-established results, the proofs of these results involve establishing the existence of density functions to study the asymptotic growth of the ideals arising from the powers of graded ideals. This talk is based on two joint works with Suprajo Das and Sudeshna Roy.

Oct. 16, 2025

Speaker

Junliang Shen (Yale)

Title

Cohomology of the universal Jacobian and compactifications

Abstract

The Jacobian variety is a fundamental object associated with a curve. The universal Jacobian over the moduli space of curves combines the geometric complexity of both abelian varieties and the moduli of curves. The purpose of this talk is to discuss two fundamental questions concerning the cohomology/Chow theory of universal Jacobians: (1) the dependence of the cup product (or intersection product) on the degree. (2) the dependence on the choice of compactification. Our approach combines the Fourier transform techniques of Beauville and Deninger–Murre, introduced over 30 years ago, with recent developments in the study of perverse filtrations associated with abelian fibrations. Based on joint work with Younghan Bae, Davesh Maulik, and Qizheng Yin.

Oct. 30, 2025

Speaker

Kangjin Han (DGIST and Notre Dame)

Title

On the ideals of cyclic Gaussian graphical models

Abstract

In this talk, we introduce a conjecture due to Sturmfels and Uhler concerning generation of the prime ideal of the variety associated to the Gaussian graphical model of any cycle graph and explain how to prove it using commutative algebra and combinatorics. Our methods are also applicable to a large class of ideals with radical initial ideals. This work is done jointly with A. Conner and M. Michalek.

Nov. 6, 2025

Speaker

Yairon Cid-Ruiz (North Carolina)

Title

Mixed zeta Segre functions and their log-concavity

Abstract

In this talk, we introduce the mixed Segre zeta function associated with a sequence of homogeneous ideals in a polynomial ring. This power series encodes information about the mixed Segre classes that arise when the ideals are extended to projective spaces of arbitrarily large dimension. Our construction generalizes and unifies classical results by Kleiman and Thorup on mixed Segre classes and by Aluffi on Segre zeta functions. We show that this function is always rational, with poles determined by the degrees of the generators of the ideals, and that it depends only on the integral closures of these ideals. Finally, we explain how the numerator of a modified form of the mixed Segre zeta function exhibits a remarkable combinatorial property: its homogenization is denormalized Lorentzian in the sense of Brändén and Huh.

Nov. 13, 2025

Speaker

Maya Banks (UIC)

Title

Degree and Syzygies of Scrolls in Weighted Projective Space

Abstract

Motivated by the relationship between the degree and the simplicity of the syzygies of projective varieties, we explore the degree of varieties embedded in weighted projective space. We construct analogs of rational normal scrolls and explore their degrees, regularity, and other syzygetic properties.

Nov. 20, 2025

Speaker

Nikola Kuzmanovski (Notre Dame)

Title

Many polynomials that are close and a few that are spread out

Abstract

In 1927, Macaulay published a paper that provides a bound on the growth of the Hilbert function of a standard graded algebra. This result has influenced many developments in algebra and combinatorics throughout the last century. In 1993, Eisenbud, Green, and Harris conjectured a generalization of Macaulay’s theorem in the presence of a regular sequence. This talk will cover some new results on the Eisenbud-Green-Harris conjecture, and some results on adjacent topics.

Nov. 24, 2025

Speaker

Uli Walther (Purdue)

Title

Resolving configuration hypersurfaces and their Nash blow up

Abstract

A configuration (the choice of a subspace W in a vector space V with distinguished basis E) over a field gives rise to a hyperplane arrangement, a matroid, and a configuration hypersurface. Bloch, Esnault and Kreimer observed that that projective configuration hypersurfaces X_W permit a natural rational surjection from a certain incidence variety. They came across it in a search for a resolution of singularities and surmised they had one. We give simple examples showing that this is not so, and exhibit a purely combinatorial property governs the smoothness of this incidence variety. Taking a closer look at the incidence variety, it also permits a second morphism, given by a "Hadamard square". We discuss that the image of this map is the Nash blow up of X_W, and that this map is the normalization map. While one can show directly that the incidence variety is normal, using Serre's criterion, we verify nice properties about its affine cone: it is in all positive characteristics F-rational. We may sketch the proof of this in the talk; it uses a duality result and a strategy that was employed before for showing that X_W and its cone are F-regular. Time permitting, we then give fewer or more details of a construction of an embedded resolution of singularities of the incidence variety (and hence also of X_W); it is based on the fact that replacing X_W with the incidence variety made it smooth in all points that lie on the torus. The main tool is theory of Tevelev that asserts a regularization process for closures (in toric varieties) of closed smooth subsets of tori. Joint with D Bath, G Denham, M Schulze.

Dec. 4, 2025

Speaker

Izzet Coskun (UIC)

Title

Realizable classes in Grassmannians

Abstract

Given a class in the cohomology of a projective manifold, one can ask whether the class can be represented by an irreducible subvariety. If the class is represented by an irreducible subvariety, we say that the class is realizable. One can further ask whether the subvariety can be taken to satisfy additional properties such as smooth, nondegenerate, rational, etc. These questions are closely related to central problems in algebraic geometry such as the Hodge Conjecture or the Hartshorne Conjecture. Recently, June Huh and collaborators have made significant progress in understanding realizable classes in products of projective spaces. In this talk, I will discuss recent joint work with Julius Ross on realizable classes in Grassmannians.

Jan. 22, 2026

Speaker

Jack Huizenga (Penn State)

Title

Brill-Noether theory for vector bundles on the projective plane

Abstract

The Brill-Noether theory of curves plays a fundamental role in the theory of curves and their moduli and has been intensively studied since the 19th century. In contrast, Brill-Noether theory for vector bundles and higher dimensional varieties is less understood. It is hard to determine when Brill-Noether loci are nonempty and these loci can be reducible and of larger than the expected dimension. In this talk, we will study Brill-Noether loci for vector bundles on the projective plane in the case where the number of sections is close to the largest possible number. When the number of sections is very large, Brill-Noether problems are all "trivial"--the Brill-Noether loci are either empty or the entire moduli space. As the number of sections decreases, we find that there is a "first" nontrivial Brill-Noether locus, and we discuss its geometry.

Jan. 29, 2026

Speaker

Fuxiang Yang (Notre Dame)

Title

Syzygies of Binary Forms and Linearly Presented Ideals

Abstract

Over the complex numbers, every binary form of degree n factors as a product of n linear forms by the fundamental theorem of algebra. Identifying the projective n-space with the space of binary forms of degree n, every partition of n determines a natural subvariety corresponding to the given factorization type. For example, the locus X_(n) where each point factors as the n-th power of a linear form is the rational normal curve, the locus X_(n-1,1) where each point factors as f^(n-1)*g is the tangent developable surface to the rational normal curve, X_(2,1,..,1) is the discriminant hypersurface, and X_(1^n) = X_(1,..,1) is the whole space. In this talk, we will discuss some recent progress on the syzygies of X_(a^b) and their connection to ideals that are linearly presented (up to finite length). The talk is in part based on joint work with Claudiu Raicu, Steven V Sam, and Jerzy Weyman.

Feb. 5, 2026

Speaker

Alessio Sammartano (Milan)

Title

The variety of orthogonal frames

Abstract

An orthogonal n-frame in a quadratic vector space of dimension d is an ordered set of n pairwise orthogonal vectors. The set of all orthogonal n-frames is an algebraic variety V(d,n). We investigate the variety V(d,n) as well as the quadratic ideal I(d,n) generated by the orthogonality relations, which cuts out V(d,n). We determine the irreducible components, classify when I(d,n) is a complete intersection or a prime ideal, and classify when the variety V(d,n) is normal or factorial. We give many applications to the theory of Lovasz-Saks-Schrijver ideals of simple graphs. If time permits, we will discuss some interesting open problems. This is a joint work with Laura Casabella.

Feb. 12, 2026

Speaker

Jakub Witaszek (Northwestern)

Title

Higher rational singularities in characteristic zero and positive characteristic

Abstract

I will begin by reviewing the theory of higher rational singularities in characteristic zero, then move on to higher F-rational singularities in positive characteristic. Finally, I will explain the idea of the proof of the deformation property of higher rational singularities. My talk will be based on joint work with Tatsuro Kawakami.

Feb. 19, 2026

Speaker

Ian Le (Australian National University)

Title

Cluster Structures on Braid Varieties

Abstract

Braid varieties are a nice family of varieties that arise in representation theory. They generalize double Bruhat cells and positroids, and are useful for studying flag varieties and Bott-Samelson varieties. In recent work with Casals, Gorsky, Gorsky, Shen and Simental, we show that a fairly general class of braid varieties admits a cluster structure. I'll explain how to construct these cluster structures, and discuss some of their combinatorial consequences.

Feb. 26, 2026

Speaker

Ritvik Ramkumar (Notre Dame)

Title

Commuting matrices and Quot schemes

Abstract

A pair of nxn matrices commute if they satisfy the obvious commuting relations (there are n^2 of them coming from the relation XY -YX = 0). But is that all? Put differently, is the variety inside C^{n^2} defined by those relations reduced and irreducible. In this talk, I will explore the long history of this problem (and its analogue for three or more matrices), connect it to Quot schemes, and discuss some (ongoing work) concerning their singularities.

Mar. 5, 2026

Speaker

Feliks Rączka (IAS)

Title

Flag varieties and the Frobenius morphism

Abstract

Let X be a smooth projective variety over an algebraically closed field of characteristic p>0 and let F:X->X be the absolute Frobenius morphism. If E is a vector bundle on X then the e-th Frobenius pushforward F^{e}_{*}E is again a vector bundle and we may try to describe its indecomposable direct summands. This problem is already interesting and non-trivial when E is the structure sheaf, and in this special case knowledge of the indecomposable direct summands of the Frobenius pushforwards carries a lot of information about the geometry of X. In this talk, I will first survey main theorems and open problems concerning this circle of ideas and then I will present my own results. For most of the talk I will focus on the case when X=G/P is a (partial) flag variety for some semi-simple algebraic group G.

Mar. 19, 2026

Speaker

Tony Varilly-Alvarado (Rice)

Title

A Bayesian Approach to the Galois action on the 27 lines on a Cubic Surface

Abstract

In 1849, Cayley and Salmon showed that every smooth complex cubic surface contains exactly 27 lines. When the surface is defined over a number field, such as Q, there is a Galois group action on these 27 lines that conjecturally detects whether the cubic surface has points with integral coordinates. This conjecture of Colliot-Thélène and Sansuc from 1979 is mediated by the Brauer group of the surface. I will explain how to use basic ideas from Bayesian inference to determine, quickly and with a high degree of confidence, the Galois action on the 27 lines of a cubic surface over Q. Time and technology permitting, I will show a live demo. This is joint work with Austen James (Rice PhD '22 -- now at Reddit).

Mar. 26, 2026

Speaker

Eric Jovinelly (Brown)

Title

Stability of Normal Bundles of General Brill-Noether Curves in P^4

Abstract

Let C be a curve of genus g. A fundamental problem in the theory of algebraic curves is to understand the embeddings of C in projective space. When the curve C is general, such an embedding is called a Brill-Noether curve if it is linearly non-degenerate. Recent work of Larson and Vogt has shown that, aside from finitely many exceptions, the normal bundle of a general Brill-Noether curve satisfies a stability property for global sections called interpolation. This leaves open the following question: do the normal bundles of general Brill-Noether curves also satisfy the more classical notion of slope stability? In this talk we will answer this question for Brill-Noether curves in four dimensional projective space: in all but a few low-genus cases the normal bundles are in fact slope stable. This is joint work with Izzet Coskun and Eric Larson.

Apr. 9, 2026

Speaker

Suhas Gondi (San Diego)

Title

Border Rank Lower Bounds for Families of GL(V)-invariant Tensors

Abstract

The border rank of tensors is a widely studied topic with practical applications to theoretical computer science and algebraic statistics. Lower bounds on the border rank of the matrix multiplication tensor were obtained using techniques from representation theory and algebraic geometry. In this talk, we will prove non-trivial border rank lower bounds for a class of GL(V)-invariant tensors using Young flattenings constructed by Wu. We will see how this comes down to proving results on ranks of certain maps between Schur functors, the proofs of which surprisingly use deep results in representation theory and commutative algebra.

Apr. 28, 2026

Speaker

Lena Ji (UIUC)

Title

Linear spaces in complete intersections of two quadrics

Abstract

In this talk, we study the linear spaces contained in the base locus of a pencil of quadrics. These encode a lot of interesting geometry: for example, for pencils of even-dimensional quadrics, there is a deep relationship between these linear spaces and hyperelliptic curves. This has found numerous applications, e.g., to rational points and to moduli spaces of vector bundles. In this talk, we focus on rationality questions for the varieties of these linear spaces, especially over non-closed fields. This work is joint with Fumiaki Suzuki.

Apr. 30, 2026

Speaker

Eamon Quinlan-Gallego (UIC)

Title

Nearby cycles and F-jumping numbers

Abstract

Given a homogeneous polynomial with isolated singularity in positive characteristic, I will show there is a connection between its F-jumping numbers and the eigenvalues of the Frobenius-fixed points of the cohomology of its "nearby fiber". I will then explain how this can be seen as a characteristic-p analogue of a much broader theorem of Kashiwara and Malgrange, and I will show how one can give an analogue of this much broader theorem by defining a nearby cycle complex in positive characteristic.