The rationality of the moduli space of genus four curves endowed with an order three subgroup of their Jacobian

TL;DR

This paper proves the rationality of a specific moduli space of genus four curves with an order three subgroup in their Jacobian by relating it to the rationality of a moduli space of points in the plane and using geometric and algebraic tools.

Contribution

It establishes the rationality of the moduli space of genus four curves with a particular Jacobian subgroup, building on cubic surface geometry and Dolgachev's theorem.

Findings

The moduli space R is birational to a tower of projective bundles.

The moduli space M of 6 points in the plane is rational.

The rationality of R follows from the rationality of M.

Abstract

Refereed version to appear in Michigan Mathematical Journal. A mistake in the last section of the previous version has been corrected. The new title exactly describes the main result obtained. Building on the geometry of cubic surfaces and on a theorem of Dolgachev, the rationality of the moduli space R mentioned in the title is proved. Let M be the moduli space of 6 points in the plane, modulo the natural involution induced by double-six configurations on cubic surfaces. It is proved that R is birational to a tower of locally trivial projective bundles ending onto M. The rationality of R then follows from Dolgachev's theorem that M is rational.