Schedule of talks:

Abstracts

(Anderson): Relations between degeneracy loci and divisors on curves go back over a hundred years: Giambelli's formula for the class of the locus where a matrix drops rank is a cornerstone of Schubert calculus; Brill and Noether's estimate for the dimension of the space of special divisors on a curve also comes from considerations of degeneracy loci. In this talk, I will describe work with Linda Chen and Nicola Tarasca which connects more recent developments in Schubert calculus with formulas by Eisenbud-Harris, Pirola, and Chan-Lopez-Pflueger-Teixidor for the genus of a Brill-Noether curve.

(Tseng): The quantum K-ring of a smooth projective variety X, introduced by Givental and YP Lee, is a deformation of the Grothendieck ring of coherent sheaves on X defined using holomorphic Euler characteristics on moduli spaces of stable maps to X (these are K-theoretic version of Gromov-Witten invariants). In this talk we discuss some properties of the quantum K-rings of X=Fl_r+1, the variety of complete flags in C^r+1, including:

(1) a presentation of the quantum K-ring;

(2) finiteness of quantum product;

(3) canonical polynomial representatives of Schubert classes.

This is based on joint work in progress with David Anderson and Linda Chen.

(Cueto): In this talk, I will discuss the structure of tropical and non-Archimedean analytic genus 2 curves and their moduli from three perspectives:

(1) as 2-to-1 covers of P¹ branched at 6 points;

(2) as solutions to the hyperelliptic equation y²=f(x), where f has degree 5; and

(3) as metric graphs dual to genus 2 nodal algebraic curves over a valued field.

Our first description allows us to give an explicit combinatorial rule to characterize each such metric graph together with a harmonic map to a metric tree on 6 leaves, in terms of the valuations of 6 branch points in P¹. Even though the tropicalization of a plane hyperelliptic curve shows no genus, we provide explicit re-embeddings of the input planar hyperelliptic curve that reveals the correct metric graphs.

From our third viewpoint, we consider the moduli space of abstract genus two tropical curves and translate the classical Igusa invariants characterizing isomorphism classes of genus two algebraic curves into the tropical realm. While these tropical Igusa functions do not yield coordinates on the tropical moduli space, we propose an alternative set of invariants that provides new length data. This is joint work with Hannah Markwig.

(Clemens): The talk will construct one example of the `equivalence' between an elliptically-fibered CY-threefold endowed with a semi-stable E<sub>8</sub> + E<sub>8</sub> bundle and a semi-stable degeneration of an elliptically-fibered CY-fourfold. I will then loosely describe how this equivalence is used to `break' the SU(5)-symmetry of grand unified theory (GUT) to the SU(3)xSU(2)xU(1)-symmetry of what physicists call the Standard Model of the physics of our cosmos.

(Baldwin): The cohomology ring of flag varieties has long been known to exhibit positivity properties. One such property is that the structure constants of the Schubert basis with respect to the cup product are non-negative. Brion (2002) and Anderson-Griffeth-Miller (2011) have shown that positivity extends to K-theory and T-equivariant K-theory, respectively. In this talk I will discuss recent work (joint with Shrawan Kumar) which generalizes these results to the case of Kac-Moody groups.

(Docampo): We study the sheaf of Kähler differentials on the arc space of an algebraic variety. We obtain explicit formulas that can be used effectively to understand the local structure of the arc space. The approach leads to new results as well as simpler and more direct proofs of some of the fundamental theorems in the literature. The main applications include: an interpretation of Mather discrepancies as embedding dimensions of certain points in the arc space, a new proof of a version of the birational transformation rule in motivic integration, and a new proof of the curve selection lemma for arc spaces. This is joint work with Tommaso de Fernex.

(Bibby): We are interested in certain arrangements of subvarieties on which a wreath product group acts. We give a combinatorial description of its poset of layers (connected components of intersections) as a generalization of Dowling and partition lattices. This combinatorial structure is an aid in understanding the cohomology of the complement as a representation of the group. In particular, when one considers a sequence of these representations, its stability can be viewed as a consequence of combinatorial stability. Joint work with Nir Gadish.

(Manon): I'll give an introduction with examples to the notion of Khovanskii basis. Khovanskii bases are generating sets for algebras which are distinguished by their computational properties as well as their relationships with tropical geometry and toric geometry. After a few examples, I'll survey some recent results on their existence and classification.

(Litt): Let X be an algebraic variety over a field k. Which representations of π₁(X) arise from geometry, e.g. as monodromy representations on the cohomology of a family of varieties over X? We study this question by analyzing the action of the Galois group of k on the fundamental group of X. As a sample application of our techniques, we show that if X is a normal variety over a field of characteristic zero, and p is a prime, then there exists an integer N=N(X,p) satisfying the following: any irreducible, non-trivial p-adic representation of the fundamental group of X, which arises from geometry, is non-trivial mod p^N.

(Gonzalez): The Cox rings of weighted projective planes blown-up at a general point have been studied in connection to the noetherianity of symbolic Rees algebras and to the finite generation of the Cox ring of the moduli space of stable rational pointed curves. We will present some sufficient conditions for the finite and nonfinite generation of the Cox ring of such blowups. This is joint work with Javier Gonzalez and Kalle Karu.

(Satriano): The Batyrev-Manin conjecture gives a prediction for the asymptotic growth of rational points on varieties over number fields when we order the points by height. The Malle conjecture predicts the asymptotic growth rate for number fields of degree d when they are ordered by discriminant. The two conjectures have the same form and it is natural to ask if they are in fact one in the same. We give a conjecture for rational point counts on stacks and show how it specializes to the two aforementioned conjectures. This is joint work with Jordan Ellenberg and David Zureick-Brown.

(Khan): One of the central questions in complex geometry is to understand the moduli space of complex structures on a given manifold. In general, this is a mysterious object. Locally, we can study deformations of complex structures using Kodaira-Spencer theory. However, the global geometry can be very complicated and have singularities. In this talk, we consider the case where we have a Riemannian-flat metric and show how this can be used to understand the geometry of the moduli space. From here, we lay out some conjectures about how this can be done more generally, and give some heuristic evidence for these conjectures.

(Ulirsch): The Hodge bundle is a vector bundle over the moduli space of smooth curves (of genus $g$) whose fiber over a smooth curve is the space of abelian differentials on this curve. We may define a tropical analogue of its projectivization as the moduli space of pairs $(\Gamma, D)$ consisting of a stable tropical curve $\Gamma$ and an effective divisor $D$ in the canonical linear system on $\Gamma$. This tropical Hodge bundle turns out to be of dimension $5g-5$, while the classical projective Hodge bundle has dimension $4g-4$. This means that not every pair $(\Gamma, D)$ in the tropical Hodge bundle arises as the tropicalization of a suitable element in the algebraic Hodge bundle. In this talk I am going to outline a comprehensive (and completely combinatorial) solution to the realizability problem, which asks us to determine the locus of points in the tropical Hodge bundle that arise as tropicalizations. Our approach is based on recent work of Bainbridge-Chen-Gendron-Grushevsky-M\”oller on compactifcations of strata of abelian differentials. Along the way, I will also develop a moduli-theoretic framework to understand the specialization of divisors to tropical curves as a natural tropicalization map in the sense of Abramovich-Caporaso-Payne. This talk is based on joint work with Bo Lin as well as with Martin Moeller and Annette Werner.

(Baker): Hyperfields are similar to their non-hyper counterparts, but addition is allowed to be multi-valued. We introduce a generalization of hyperfields called tracts, which also include partial fields in the sense of Semple and Whittle. After giving some examples, we present a simultaneous generalization of the notion of linear subspaces, matroids, valuated matroids, oriented matroids, and regular matroids. We call such objects matroids over tracts. In fact, there are (at least) two natural notions of matroids over a tract F, which we call weak and strong F-matroids. We give different ``cryptomorphic'' axiom systems for such matroids, discuss duality theory, and present sufficient conditions which guarantee that the notions of weak and strong F-matroids coincide (for a given tract F).

(Tarasca): In this talk, I will discuss results on the enumerative geometry of subvarieties of moduli spaces of curves. I will present a formula for effective classes of loci of genus-two stable curves with marked Weierstrass points. The formula is expressed as a sum over stable graphs indexing boundary strata of moduli spaces of pointed curves. This result follows by analyzing the interplay of geometry and combinatorics of curves and their moduli. This is a first step in the study of the structure of hyperelliptic classes in all genera. Joint work with Renzo Cavalieri.

(Fulton): In the 1840's, Cayley and Salmon found formulas for the loci of m by n matrices which do not have maximal rank. Because these varieties are defined by many more equations than their codimension, finding such formulas has been a challenge and stimulus to the development of intersection theory. This talk will sketch some of this history, which continues today.

(Hofscheier): A lattice polytope is called spanning if its lattice points affinely span the ambient lattice. In this talk, I will present recent joint work with Lukas Katthan and Benjamin Nill, where we generalize Harris' Uniform Position Principle to obtain new inequalities for the h*-vector of spanning lattice polytopes. This extends Hibi's inequality for polytopes with interior lattice points, as well as certain inequalities due to Stanley.

(Brion): The objects of the talk are the translation invariant vector bundles on an abelian variety. They form a tensor abelian category which is equivalent to the category of coherent sheaves with finite support on the dual abelian variety, via the Fourier-Mukai transform. The talk will present an alternative approach to the category of homogeneous vector bundles, in terms of represen- tations of a commutative affine group scheme. Some natural operations on vector bundles such as tensor product, dual, push-forward and pull-back under isogenies, can be described in simple terms via this approach.

Past Seminars

Ohio State University Algebraic Geometry Seminar-Year 2016-2017

Ohio State University Algebraic Geometry Seminar-Year 2015-2016

Ohio State University Algebraic Geometry Seminar-Year 2014-2015

Ohio State University Algebraic Geometry Seminar-Year 2013-2014

This page is maintained by Angie Cueto and Dave Anderson.