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

In this note we look at the moduli space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal R}_{3,2}$\end{document} 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 \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal R}_{3,2} \rightarrow {\mathcal A}_4$\end{document} to Siegel space. We show that there is a birational model of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal R}_{3,2}$\end{document} as a group quotient of a product of two Grassmannian varieties. This gives a proof of the unirationality of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal R}_{3,2}$\end{document} and hence a new proof for the unirationality of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal A}_4$\end{document}.