Schedule of talks:

See also the Arithmetic Geometry Seminar

Abstracts

(Haburcak): Classical Brill-Noether theory studies linear systems on a general curve in the moduli space M_g of algebraic curves of genus g. A refined Brill-Noether theory studies the linear systems on curves with a given Brill-Noether special linear system. As a first step, one would like to understand the stratification of M_g by Brill-Noether loci, which parameterize curves with a particular projective embedding. We'll discuss recent progress on distinguishing Brill-Noether loci and the recent resolution of the Maximal Brill-Noether loci conjecture, identifying the maximal elements of the stratification by Brill-Noether loci. This is based on joint work with Asher Auel, Andrei Bud, Andreas Leopold Knutsen, and Hannah Larson.

(Newman): One can use Bloch's higher Chow groups to compute the usual Chow groups of moduli spaces. This involves first computing the necessary higher Chow groups, and then computing the connecting homomorphism of the localization exact sequence. I will explain general techniques for performing these computations and give examples for the integral Chow groups of moduli spaces of genus 1 curves.

(Coles): In recent decades tropical geometry, the study of algebraic varieties by constructing an associated 'combinatorial shadow', has proven to be a fruitful toolkit in many areas of algebraic geometry. The starting point for constructing these shadows has often been the tropicalization of toric varieties. In this talk I will explain how tropicalization can be extended to a larger class of varieties, spherical varieties, and I will also discuss the connection with Berkovich geometry.

(Tseng): Hilbert schemes of points on the affine plane are arguably the moduli spaces of sheaves with the richest known structures. The Deligne-Mumford moduli spaces of stable curves are by far the most studied moduli spaces of varieties in algebraic geometry. Stable maps provide a system of correspondences between these moduli spaces. In particular, one can construct (cycle) classes on moduli spaces of curves from classes on Hilbert schemes. In this talk, we discuss a numerical study of these classes in the genus 1 case.

(Setiabrata): Double Grothendieck polynomials G_w(x;y) are lifts of structure sheaves of Schubert varieties in the equivariant K-theory of the flag variety. Motivated by our search for a representation-theoretic avatar for these polynomials, we give a new formula for G_w(x;y) based on Magyar's orthodontia algorithm for diagrams. We obtain a similar formula for double Schubert polynomials S_w(x;y), and a curious positivity result: For vexillary permutations w, the polynomial x_1^n \dots x_n^n S_w(x_n^{-1}, \dots, x_1^{-1}; 1, \dots, 1) is a graded nonnegative sum of Lascoux polynomials. This is joint work with Avery St. Dizier.

(Lee): The aim of this talk is to explain that on a Calabi-Yau threefold a genus zero quantum K-invariant can be written as an integral linear combination of a finite number of Gopakumar--Vafa BPS invariants with coefficients from an explicit multiple cover formula. Conversely, all Gopakumar-Vafa invariants can be determined by a finite number of quantum K-invariants in a similar manner. The technical heart is a proof of a remarkable conjecture by Hans Jockers and Peter Mayr. This result is consistent with the ''virtual Clemens conjecture'' for the Calabi-Yau threefolds, a weak version of which has been proved by John Parden in cohomology (but not in K-theory). This is a joint work with You-Cheng Chou.

(Cavey): The Hilbert schemes of points on a smooth algebraic surface are smooth varieties that parametrize finite closed subschemes of the surface of a fixed length. When the underlying surface is toric, global sections of line bundles on the surface correspond to integer points in an associated polygon. In this talk, I will explain how in certain cases the corresponding problem on the Hilbert scheme can be interpreted as a packing problem for integer points in the same polygon satisfying a certain separation condition. Such an interpretation is known for all ample line bundles on Hilbert schemes of points on Hirzebruch surfaces (for example P¹ x P¹) and is expected to hold more generally. Based on this counting interpretation for sections of ample line bundles, I will also give formula for the Euler characteristic of any line bundle on the Hilbert schemes of points on P¹ x P¹. The latter formula has applications to the Verlinde series introduced by Ellingsrud, Göttsche, and Lehn.

(Moretti): Given a projective variety, one can understand rational maps to a projective space of the same dimension in terms of the associated kernel bundle. In the case of K3 surfaces of Picard rank 1, this allows to prove that rational maps of degree at most d (induced by the primitive linear system) move in families. I will explain how, by combining vector bundle techniques with derived category methods, one can study and characterize in many cases rational maps of minimal degree for polarized K3 surfaces of genus up to 14. This can be seen as a preliminary step towards computing the degree of irrationality of these surfaces. Joint work with Andrés Rojas.

Past Seminars

Ohio State University Algebraic Geometry Seminar-Year 2023-2024

Ohio State University Algebraic Geometry Seminar-Year 2022-2023

Ohio State University Algebraic Geometry Seminar-Year 2021-2022

Ohio State University Algebraic Geometry Seminar-Year 2020-2021

Ohio State University Algebraic Geometry Seminar-Year 2019-2020

Ohio State University Algebraic Geometry Seminar-Year 2018-2019

Ohio State University Algebraic Geometry Seminar-Year 2017-2018

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

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