# Poster Abstracts

Richard Haburcak

**Title: **Maximal Brill-Noether loci via K3 Surfaces

**Abstract:** The Brill-Noether (BN) loci $\mathcal{M}^r_{g,d}$ parameterize BN special curves of genus g admitting a line bundle of degree d and r+1 global sections. We explain a strategy for distinguishing BN loci by studying the lifting of line bundles on curves on polarized K3 surfaces and identify the maximal BN loci in genus 9-19, 22, and 23. Specifically, we prove new results concerning when a line bundle of type g^{3}_{d} on a curve is a restriction of a line bundle on the K3 surface via an analysis of the stability of Lazarsfeld-Mukai bundles.

Brian Hepler

**Title:** Moderate Growth and Rapid Decay Nearby Cycles using Enhanced Ind-Sheaves

**Abstract:** We define moderate growth and rapid decay objects associated to any enhanced ind-sheaf on a complex manifold X of arbitrary dimension and a holomorphic function f on X. We show that these objects recover in particular---via the irregular Riemann-Hilbert correspondence of D'Agnolo-Kashiwara---the moderate growth and rapid decay de Rham complexes of a holonomic D-module. This also enables us to deduce well-known duality pairings by topological means. Moreover, our new objects admit natural distinguished triangles, generalizing those known from the classical theory of nearby and vanishing cycles for constructible sheaves. This is joint work with Andreas Hohl at IMJ-PRG.

Colette LaPointe

**Title:** Some dynatomic modular curves in positive characteristic

**Abstract:** In arithmetic dynamics, $f_c(x)=x^d+c$ has been the most frequently studied one-dimensional polynomial family in any characteristic. It has been shown by J.R. Doyle et al. (2019) for example that if $p$ is a prime not dividing $d$ but dividing $n>3$, then the dynatomic modular curve $Y_1(n)$, which parametrizes the maps $f_c$ with a marked point of period $n$, has bad reduction modulo $p$. Less is known about $Y_1(n)$ or related dynatomic curves like $Y_0(n)$, which parametrize the maps $f_c$ with a marked orbit of period $n$, in positive characteristic for other polynomial families. In the current research we have shown that over characteristic $p$, the curve $Y_1(n)$ for the family $f_c(x)=x^p+x+c$ is smooth if and only if $p\nmid n$, except in the case $n=p=2$ where $Y_1(n)$ is smooth. $Y_1(n)$ has also been shown to be non-reduced when $n=p^k$ for $k\geq1$ (except in the case $n=p=2$), and from computation on Sage, we know $Y_1(n)$ can be reducible even in some cases when $Y_1(n)$ is smooth. Some further questions now being explored include finding when $Y_1(n)$ irreducible, and what can be said on the smoothness or irreducibility of $Y_0(n)$.

Jack Petok

**Title:** Sections of quadric fourfold bundles

**Abstract:** Given a quadric fourfold bundle over the projective plane with section, there is construction originating in the classical theory of quadratic forms (reduction by hyperbolic splitting), which produces an associated quadric surface bundle. In an ongoing project with Asher Auel, we use hyperbolic reduction to relate two different parameter spaces: the moduli space of sections of a fixed height and the moduli space of twisted sheaves on a K3 surface.

Weite Pi

**Title:** Generators for the cohomology ring of the moduli of one-dimensional sheaves on P^2

**Abstract:** We explore the structure of the cohomology ring of the moduli space of stable 1-dimensional sheaves on ℙ2 of any degree. We obtain a minimal set of tautological generators, which implies an optimal generation result for both the cohomology and the Chow ring of the moduli space. Our approach is through a geometric study of tautological relations. This is joint work with Junliang Shen. Preprint available at https://arxiv.org/abs/2204.05866.

Geoffrey Smith

**Title:** Stability and cohomology of kernel bundles on projective space

Weihong Xu

**Title:** Quantum K-theory of Incidence Varieties

**Abstract:** Buch, Chaput, Mihalcea, and Perrin proved that for cominuscule flag varieties, (T-equivariant) K-theoretic (3-pointed, genus 0) Gromov-Witten invariants can be computed in the (equivariant) ordinary K-theory ring. Buch and Mihalcea have a related conjecture for all type A flag varieties. I will discuss work that proves this conjecture in the first non-cominuscule case--the incidence variety X=Fl(1,n-1;n). The proof is based on showing that Gromov-Witten varieties of stable maps to X with markings sent to a Schubert variety, a Schubert divisor, and a point are rationally connected. As applications, I will discuss positive formulas (an equivariant Chevalley formula and a non-equivariant Littlewood-Richardson rule) which determine the (equivariant) quantum K-theory ring of X. The Littlewood-Richardson rule implies that non-empty Gromov-Witten varieties given by Schubert varieties in general position have arithmetic genus 0. The poster is based on the arxiv preprint at https://arxiv.org/abs/2112.13036.