The Chow rings of the moduli spaces of curves of genus 7, 8, and 9
In recent joint work with Hannah Larson, we determine the Chow rings of the moduli spaces of curves of genus 7, 8, and 9 by showing that they are generated by tautological classes and using Faber's computation of the tautological ring. I will explain what it means to be tautological and the key properties of the moduli spaces of low genus curves that make it tractable to compute their Chow rings.
The geography of negative curves
We'll talk about the Mori dream space (MDS) property for blowups of weighted projective planes (WPP) at a general point. Such a variety is a MDS if and only if it contains a non-exceptional negative curve and another curve disjoint from it.
From a toric perspective a WPP is defined by a rational plane triangle. We'll consider a parameter space of all such triangles and see how the negative curves and the MDS property vary within it. Our main goal will be to explain how the knowledge of negative curves in this parameter space often determines the MDS property. This is joint work with José Luis González and Kalle Karu.
Wall crossing morphisms for moduli of stable pairs
Consider a quasi-compact moduli space M of pairs (X,D) consisting of a variety X and a divisor D on X. If M is not proper, it is reasonable to find a compactification of it. Assume furthermore that there are two rational numbers 0<b<a<1 such that, for every pair (X,D) corresponding to a point in M, the pairs (X,aD) and (X,bD) are klt, and the Q-divisors K_X+aD and K_X+bD are ample. Using Kollár's formalism of stable pairs, one can construct two different compactifications of M (M_a and M_b), corresponding to a and b. One may wonder how to relate these two compactifications. The main result is that, up to replacing M_a and M_b with their normalizations, there are birational morphisms M_a \to M_b. This project is inspired by Hassett's work on weighted stable curves, and is joint with Kenny Ascher, Dori Bejleri and Zsolt Patakfalvi.
The Noether–Lefschetz theorem
The classical Noether–Lefschetz theorem says that for a very general surface $S$ of degree $\geq 4$ in $\mathbb P^3$ over the complex numbers, the restriction map from the divisor class group on $\mathbb P^3$ to $S$ is an isomorphism. In this talk, we give an elementary proof of Noether–Lefschetz. We do not use any Hodge theory, cohomology, or monodromy. This argument has the additional advantage that it works over fields of arbitrary characteristic and for singular varieties (for Weil divisors).
Stable Trace Ideals and Arf Rings
In this talk we will explore the intersection of trace ideals and stable ideals, that is, ideals that are stable under homomorphisms to the ring and ideals that are isomorphic to their endomorphism rings. We apply our results to the study of Arf rings. This is ongoing joint work with Hailong Dao.
A tropical approach to the enriched count of bitangents to quartic curves
Using A1 enumerative geometry Larson and Vogt have provided an enriched count of the 28 bitangents to a quartic curve. In this talk, I will explain how these enriched counts can be computed combinatorially using tropical geometry. I will also introduce an arithmetic analogue of Viro’s combinatorial patchworking for real algebraic curves, which in some cases retains enough data to recover the enriched counts. Finally, I will comment on a possible tropical approach to the enriched count of the 27 lines on a cubic surface of Kass and Wickelgren. This talk is based on joint work with Hannah Markwig and Sam Payne.
The Equivariant Ehrhart Theory of the Permutahedron
In 2010, motivated by representation theory, Stapledon described a generalization of Ehrhart theory with group actions. In 2018, Ardila, Schindler, and I made progress towards answering one of Stapledon’s open problems that asked to determine the equivariant Ehrhart theory of the permutahedron. We proved some general results about the fixed polytopes of the permutahedron, which are the polytopes that are fixed by acting on the permutahedron by a permutation. In particular, we computed their dimension, showed that they are combinatorially equivalent to permutahedra, provided hyperplane and vertex descriptions, and proved that they are zonotopes. Lastly, we obtained a formula for the volume of these fixed polytopes, which is a generalization of Richard Stanley’s result of the volume for the standard permutahedron. Building off of the work of the aforementioned, Ardila, Supina, and I determine the equivariant Ehrhart theory of the permutahedron, thereby resolving the open problem. This project presents formulas for the Ehrhart quasipolynomials and Ehrhart Series of the fixed polytopes of the permutahedron, along with other results regarding interpretations of the equivariant analogue of the Ehrhart series. This is joint work with Federico Ardila (San Francisco State University, Anna Schindler (North Seattle College) and Mariel Supina (UC Berkeley).
Odd degree isolated points on X_1(N) with rational j-invariant
For a curve C defined over a number field K, we say that a closed point x in C of degree d is isolated if it does not belong to an infinite family of degree d points paramaterized by either the projective line or a positive rank abelian subvariety of the curve's Jacobian. In this talk, we characterize elliptic curves with rational j-invariant that give rise to an isolated point of odd degree on X_1(N)/Q for some positive integer N. Joint with Abbey Bourdon, David R. Gill, and Jeremy Rouse.