I will report on a joint work in progress with I. Biswas, E. Colombo and A. Ghigi in which we describe a canonical projective structure on every etale double cover of a curve $C$ of genus $g>6$. This projective structure is the restriction to the second infinitesimal neighborhood of the diagonal in $C\times C$ of the second fundamental form of the Prym map. It gives a section of the space of projective structures on $\mathbb{R}_g$ and the $(0,1)$-component of the differential of this section is proven to be the pullback via the Prym map of the Kaehler form on $A_{g-1}$. This generalises a previous result obtained in collaboration with Biswas, Colombo, and Pirola in the case of $M_g$, showing the existence of a canonical projective structure on every curve of genus $g>3$, obtained by the second fundamental form of the Torelli map.