Fibrations in abelian varieties associated to Enriques surfaces

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 C → |C| and D → f*|C| ⊂ |D| be the universal family and the restriction of the universal family respectively. We construct the relative Prym variety P = Prym( D,C ) of D over C 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 chi = d - g + 1 we consider the degree d relative compactified Jacobian N = JacdA (|C|) → |C|, with respect to a polarization A on T. If chi is such that the Mukai vector (0, [D], 2chi) 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) ≅ Z (2), that oN ≅ ON , and that hp ,0(N) = 0 for p ≠ 0, 2 g - 1.