Some singular curves in Mukai's model of $\overline{M}_7$

TL;DR

This paper explores singular curves within Mukai's model of the moduli space of genus 7 curves, using computational tools to identify and analyze specific boundary objects and their stability.

Contribution

It introduces the first explicit examples of singular curves in Mukai's genus 7 model and examines their stability using computational invariant theory.

Findings

Identified three types of singular curves: a 7-cuspidal curve, a balanced ribbon, and reducible nodal curves.

Established Spin(10)-semistability for these singular curves via invariant polynomial evaluation.

Provided computational methods for analyzing boundary objects in Mukai's model.

Abstract

Mukai showed that the GIT quotient $\operatorname{Gr}(7,16) /\!/ \operatorname{Spin}(10)$ is a birational model of the moduli space of Deligne-Mumford stable genus 7 curves $\overline{M}_7$. The key observation is that a general smooth genus 7 curve can be realized as the intersection of the orthogonal Grassmannian $\operatorname{OG}(5,10)$ in $\mathbb{P}^{15}$ with a six-dimensional projective linear subspace. What objects appear on the boundary of Mukai's model? As a first step in this study, computer calculations in Macaulay2, Magma, and Sage are used to find and analyze linear spaces yielding three examples of singular curves: a 7-cuspidal curve, the balanced ribbon of genus 7, and a family of genus 7 reducible nodal curves. $\operatorname{Spin}(10)$-semistability is established by constructing and evaluating an invariant polynomial.