Stability of genus five canonical curves

TL;DR

This paper investigates the geometric invariant theory stability of nets of quadrics in projective 4-space, linking it to the moduli space of genus 5 curves and the minimal model program.

Contribution

It establishes the GIT stability analysis for nets of quadrics and connects the resulting quotient to the minimal model program for genus 5 curves.

Findings

GIT stability of nets of quadrics is characterized.

The GIT quotient models the moduli space of genus 5 curves.

The quotient corresponds to the final step in the log minimal model program.

Abstract

We analyze GIT stability of nets of quadrics in $\mathbb{P}^4$ up to projective equivalence. Since a general net of quadrics defines a canonically embedded smooth curve of genus five, the resulting GIT quotient gives a birational model of the moduli space of genus 5 curves. We study the geometry of the associated contraction and prove that the constructed GIT quotient is the final step of the log minimal model program for the moduli space of genus 5 curves.