Prym varieties and projective structures on Riemann surfaces

Given an \'etale double covering $\pi\, :\, \widetilde{C}\, \longrightarrow\, C$ 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 $\widetilde C$ and $C$. This construction can be interpreted as a section of an affine bundle over the moduli space of \'etale double covers. The $\overline{\partial}$--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.