Browsing by Author "Sacca, Giulia"

Let $T$ be a general Enriques surface and let $f:S\to T$ be its universal cover. Consider a smooth curve $C\subset T$ of genus $g\ge 2$, set $D:=f-1(C)$, and let $\backslash mcC\to |C|$ and $\backslash mcD\to f\ast |C|\subset |D|$ be the universal family and the restriction of the universal family respectively. We construct the relative Prym variety $P=\backslash Prym(\backslash mcD,\backslash mcC)$ of $\backslash mcD$ over $\backslash mcC$ and show that it is a (possibly singular) symplectic variety of dimension $2g-2$. There is a morphism $P\to |C|$, which is a Lagrangian fibration and whose smooth fibers are $(g-1)$-dimensional Prym varieties. We also prove that the smooth locus of $P$ is simply connected. For any non zero integer $\chi =d-g+1$ we consider the degree $d$ relative compactified Jacobian $N=\backslash JacAd(|C|)\to |C|$, with respect to a polarization $A$ on $T$. If $\chi$ is such that the Mukai vector $(0,[D],2\chi )$ is primitive in $H\ast (S,Z)$, and $A$ if is general, we prove that $N$ is smooth. Moreover, under some technical assumption that can be verified for low values of $g$ and that are expected to be true in general, we show that $\pi 1(N)\cong \backslash Z/(2)$, that $\omega N\cong \backslash mcON$, and that $hp,0(N)=0$ for $p\ne 0,2g-1$.