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

Abstract

In this note we look at the moduli space R_3,2 of double covers of genus three curves, branched along 4 distinct points. This space was studied by Bardelli, Ciliberto and Verra in 1. It admits a dominating morphism R_3,2 → A₄ to Siegel space. We show that there is a birational model of R_3,2 as a group quotient of a product of two Grassmannian varieties. This gives a proof of the unirationality of R_3,2 and hence a new proof for the unirationality of A₄.