Geometry of Prym semicanonical pencils and an application to cubic threefolds

Abstract

In the moduli space scriptRg$\mathcal {R}_g$ of double étale covers of curves of a fixed genus g, the locus formed by covers of curves with a semicanonical pencil consists of two irreducible divisors scriptTge$\mathcal {T}^e_g$ and scriptTgo$\mathcal {T}^o_g$. We study the Prym map on these divisors, which shows significant differences between them and has a rich geometry in the cases of low genus. In particular, the analysis of scriptT5o$\mathcal {T}^o_5$ has enumerative consequences for lines on cubic threefolds.