Ohio State University Algebraic Geometry Seminar


Ohio State University Algebraic Geometry Seminar Year 2025-2026 Time: Tuesdays 10:20-11:15am Location: MW 154 (in person) or Zoom (virtual, email the organizers for the Zoom coordinates)

Schedule of talks:


TIME	SPEAKER	TITLE
August 26 Tue, 10:20am	Adrian Neff (OSU)	Residues and contractions of genus one curves
September 9 Tue, 10:20am	Yiyu Wang (OSU)	The local Euler obstruction of a matroid Schubert variety
September 23 Tue, 10:20am	Amy Li (Texas)	Intersection theory on the Hurwitz space of admissible covers
September 30 Tue, 10:20am	Jingxiang Ma (Sheffield)	Crepant resolutions and curve counting for type D du Val singularities
October 7 Tue, 10:20am	Hsian-Hua Tseng (OSU)	More on elliptic curves and Hilbert schemes
October 21 Tue, 10:20am	Will Newman (OSU)	Injective Quadratic Self-Maps of $\mathbb{P}^2$
October 28 Tue, 10:20am	Patricia Klein (Texas A&M)	Introduction to geometric vertex decomposition
November 18 Tue, 10:20am	Juan Pablo Zuñiga (Pontificia Universidad Católica de Chile)
December 2 Tue, 10:20am	Martin Bishop (Northwestern)
January 13 Tue, 10:20am	( )
January 20-22 4pm	June Huh (Princeton)	Zassenhaus lectures

See also the Arithmetic Geometry Seminar

Abstracts

(Neff): Given a nodal genus one curve and a proper subcurve of genus one, we will construct a contraction that collapses the subcurve to a genus one singularity. We will do this by first introducing residues for curves over local artinian rings and "twists" of these residues by tropical data from the curve, then we will use these residues to explicitly construct the contraction.

(Wang): Matroid Schubert varieties--Schubert varieties of hyperplane arrangements--serve a role in the Kazhdan-Lusztig theory of matroids analogous to that of classical Schubert varieties in geometry. These varieties carry rich combinatorial data about the underlying hyperplane arrangements. The local Euler obstruction, a key invariant introduced by MacPherson in his definition of Chern classes for singular varieties, captures local information of the singularity. In this talk, we study the local Euler obstruction of matroid Schubert varieties and demonstrate that it is a combinatorial invariant: specifically, it equals the evaluation of the characteristic polynomial at 2.

(Li): The Hurwitz space is a moduli space parametrizing branched covers of curves. Harris and Mumford introduced the "admissible covers" compactification of the Hurwitz space, in which the target and source curves of a cover degenerate into nodal curves when branch points come together. The boundary of the Hurwitz space is then stratified by lower-dimensional Hurwitz spaces. This structure is strikingly similar to the stratification of the Deligne-Mumford compactification of the moduli space of curves. Inspired by the well-studied intersection theory of the moduli space of curves, we develop new techniques for computing the low degree Chow and cohomology groups of the Hurwitz space. Some of this work is joint with E. Clader, Z. Hu, H. Larson, and R. Lopez.

(Ma): A resolution of singularities $f: Y\to X$ is crepant if it preserves the canonical class, i.e. $f^*K_X = K_Y$. The crepant resolution conjecture predicts deep and nontrivial relations between the Gromov-Witten invariants of $X$ and those of $Y$. These invariants, which can be thought of as counting curves inside the variety, play an important role in both mathematics and physics. In this talk, I will describe ongoing work on this conjecture for du Val singularities of type D---a class of examples lying beyond the toric case. A key new ingredient is recent work on mirror symmetry for the minimal resolutions of du Val singularities joint with Andrea Brini and Ian Strachan.

(Tseng): We will discuss an ongoing project whose goal is to explicitly solve the genus 1 Gromov-Witten theory of Hilbert schemes of points on a smooth (quais)projective surface. This is joint with Rahul Pandharipande.

(Newman): Given a rational map from $\mathbb{P}^n$ to $\mathbb{P}^n$ defined over a field k, one can ask when it is injective on k-rational points. For maps defined by quadratic polynomials, we relate this injectivity to the existence of rational points of ramification, which gives a complete characterization when n=2. When k is finite, we see that this characterization severely restricts the geometry of the map. From this, we can write down explicit equations on the coefficients that determine injectivity. This is joint work with Michael Zieve.

(Klein): Vertex decomposition, introduced by Provan and Billera in 1980, is an inductive strategy for breaking down and understanding simplicial complexes. A simplicial complex that is vertex decomposable is shellable, hence Cohen--Macaulay. Through the Stanley--Reisner correspondence, vertex decompositions of simplicial complexes inform the study of squarefree monomial ideals, which we will be interested in studying in their capacity as the defining ideals of initial varieties of combinatorially-defined varieties. One limitation of this story is that a great deal of the combinatorial information defining a variety may be lost when one jumps in one fell swoop to its initial scheme. A generalization of vertex decomposition, called geometric vertex decomposition, introduced by Knutson, Miller, and Yong in 2009, allows one to take smaller steps towards the initial scheme, preserving more of the original combinatorial information along the way. In this talk, we will describe and give examples of geometric vertex decompositions in combinatorially-natural settings and describe some inferences that geometric vertex decomposition allows us to make.

(Zuñiga):

(Bishop):