Fibrations in abelian varieties associated to Enriques surfaces

Abstract

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 $\mc C \to |C|$ and $\mc D \to f^*|C|\subset |D|$ be the universal family and the restriction of the universal family respectively. We construct the relative Prym variety $P=\Prym(\mc D , \mc C)$ of $\mc D$ over $\mc C$ 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 $N=\Jac^d_A(|C|) \to |C|$, with respect to a polarization $A$ on $T$. If $\chi$ is such that the Mukai vector $(0,[D],2\chi)$ is primitive in $H^*(S,Z)$, 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 $\pi_1(N)\cong \Z/(2)$, that $\omega_N\cong {\mc O}_N$, and that $h^{p,0}(N)=0$ for $p\neq 0, 2g-1$.