A note on the unirationality of a moduli space of double covers

TL;DR

This paper proves the unirationality of a moduli space of double covers of genus three curves, providing a new approach via a birational model as a group quotient of Grassmannians, which also implies the unirationality of ${ m A}_4$.

Contribution

It constructs a birational model of the moduli space as a group quotient of Grassmannians, establishing its unirationality and offering a new proof for the unirationality of ${ m A}_4$.

Findings

The moduli space $ R_{3,2}$ is unirational.

A birational model of $ R_{3,2}$ as a group quotient of Grassmannians is constructed.

Unirationality of ${ m A}_4$ is derived from the model.

Abstract

In this note we look at the moduli space $\cR_{3,2}$ of double covers of genus three curves, branched along 4 distinct points. This space was studied by Bardelli, Ciliberto and Verra. It admits a dominating morphism $\cR_{3,2} \to {\mathcal A}_4$ to Siegel space. We show that there is a birational model of $\cR_{3,2}$ as a group quotient of a product of two Grassmannian varieties. This gives a proof of the unirationality of $\cR_{3,2}$ and hence a new proof for the unirationality of ${\mathcal A}_4$.