Fibrations in abelian varieties associated to Enriques surfaces

Abstract

Let T be a general Enriques surface and let f:S→T be its universal cover. Consider a smooth curve C⊂T of genus g≥2, set D:=f−1(C), and let \mcC→|C| and \mcD→f∗|C|⊂|D| be the universal family and the restriction of the universal family respectively. We construct the relative Prym variety P=\Prym(\mcD,\mcC) of \mcD over \mcC and show that it is a (possibly singular) symplectic variety of dimension 2g−2. There is a morphism P→|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 χ=d−g+1 we consider the degree d relative compactified Jacobian N=\JacAd(|C|)→|C|, with respect to a polarization A on T. If χ is such that the Mukai vector (0,[D],2χ) 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 π1(N)≅\Z/(2), that ωN≅\mcON, and that hp,0(N)=0 for p≠0,2g−1.