Moduli of curves of genus 6 and K-stability

Abstract: In this talk, I will describe a way to study moduli of curves of small genus (eg. $g=3,4,6$) via $K$-stability. For instance, a general curve $C$ of genus $6$ can be embedded into the unique quintic del Pezzo surface $X_5$ as a divisor of class $-2K_{X_5}$. Thus the $K$-moduli spaces of the pair $(X_5, cC)$ are birational to the moduli of DM-stable curves $\bar{M}_6$. On the other hand, $X_5$ can be embedded in $\mathbb{P}^1 \times\mathbb{P}^2$ as a divisor of class $\mathcal{O}(1,2)$, under which $-2K_X$ is linearly equivalent to $\mathcal{O}_X(2,2)$. One can study the VGIT-moduli spaces in this setting. In this talk, I will compare these various compactifications of moduli spaces.

algebraic geometrycombinatorics

Audience: researchers in the topic

Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html