The Igusa Quartic and the Prym Map, with Some Rational Moduli

This paper is devoted to the ubiquity of the Igusa quartic B subset of P-4 in connection to the Prym map p : R-6 -> A(5). We introduce the moduli space chi of those quartic threefolds X cutting twice a quadratic section of B. A general X is 30-nodal and the intermediate Jacobian J(X) of its natural desingularization is a five-dimensional p.p. abelian variety. Let j : chi -> A(5) be the period map sending X to J(X), in the paper we study j and its relation to p. As is well known the degree of p is 27 and its monodromy group endows any smooth fibre F of p with the incidence configuration of 27 lines of a cubic surface. Then the same monodromy defines a map j' : D-6 -> A(5) of degree 36, with fibre the configuration of 36 'double-six' sets of lines of a cubic surface. We prove that j = j' omicron phi, where phi : chi -> D-6 is birational. This provides a geometric description of j'. Finally we describe the relations between the different moduli spaces considered and prove that some, including chi, are rational.