New symplectic V-manifolds of dimension four via the relative compactified Prymian

Three new examples of 4-dimensional irreducible symplectic V-manifolds are constructed. Two of them are relative compactified Prymians of a family of genus-3 curves with involution, and the third one is obtained from a Prymian by Mukai's flop. They have the same singularities as two of Fujiki's examples, namely, 28 isolated singular points analytically equivalent to the Veronese cone of degree 8, but a different Euler number. The family of curves used in this construction forms a linear system on a K3 surface with involution. The structure morphism of both Prymians to the base of the family is a Lagrangian fibration in abelian surfaces with polarization of type (1,2). No example of such fibration is known on nonsingular irreducible symplectic varieties.Comment: 28 page

A parametrization of the theta divisor of the quartic double solid

Let MX(2; 0, 3) be the moduli space of rank-2 stable vector bundles with Chern classes c1 = 0, c2 = 3 on the Fano threefold X, the double solid P 3 of index two. We prove that the vector bundles obtained by Serre’s construction from smooth elliptic quintic curves on X form an open part of an irreducible component M of MX(2; 0, 3) and that the Abel-Jacobi map φ: M → J(X) into the intermediate Jacobian J(X) of X defined by the second Chern class is generically finite of degree 84 onto a translate Θ + const of the theta-divisor. We also prove that the family of elliptic quintics on a general X is irreducible and of dimension 10