Geometry of Prym semicanonical pencils and an application to cubic threefolds

In the moduli space $\mathcal{R}_g$ of double \'etale covers of curves of a fixed genus $g$, the locus formed by covers of curves with a semicanonical pencil consists of two irreducible divisors $\mathcal T^e_g$ and $\mathcal T^o_g$. 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 $\mathcal T^o_5$ has enumerative consequences for lines on cubic threefolds.