Prym varieties and projective structures on Riemann surfaces

Given an étale double covering $\pi :\tilde{C}⟶C$ of compact Riemannsurfaces with $C$ of genus at least two, we use the Prym variety of the cover to construct canonical projective structures on both $\tilde{C}$ and $C$. This construction can be interpreted as a section of an affine bundle over the moduli space of étale double covers. The $\bar{\mathrm{\partial }}$–derivative of this section is a (1,1)–form on the moduli space. We compute this derivative in terms of Thetanullwert maps. Using the Schottky–Jung identities we show that, in general, the projective structure on $C$ depends on the cover.