The rationality of certain moduli spaces of curves of genus 3

TL;DR

This paper proves that certain moduli spaces of genus 3 curves with specific torsion line bundles are rational over any algebraically closed field, advancing understanding of their geometric structure.

Contribution

It establishes the rationality of two particular moduli spaces of genus 3 curves with 3-torsion divisor classes over any algebraically closed field.

Findings

The moduli space M(3,3) is rational over any algebraically closed field.

The moduli space M(3,<3>) is rational over any algebraically closed field.

Abstract

We show, for each algebraically closed field, the rationality of the following two moduli spaces: M(3,3) parametrizing pairs (C, \eta) where C has genus 3 and \eta is a 3-torsion divisor class, respectively of M(3,<3>) parametrizing pairs (C, <\eta>) as above and where <\eta> is the cyclic subgroup of order 3 in Pic_0(C) generated by \eta.