Moduli of non-standard Nikulin surfaces in low genus

Abstract

Primitively polarized genus g Nikulin surfaces (S, M, H) are of two types, that we call standard and non-standard depending on whether the lattice embedding Z[H] ⊕⊥ N ⊂ Pic S is primitive. Here H is the genus g polarization and N is the Nikulin lattice. We concentrate on the non-standard case, which only occurs in odd genus. In particular, we study the birational geometry of the moduli space of nonstandard Nikulin surfaces of genus g and prove its rationality for g = 7, 11 and the existence of a rational double cover of it when g = 9. Furthermore, if (S, M, H) is general in the above moduli space and (C, M|C ) is a general Prym curve in |H|, we determine the dimension of the family of non-standard Nikulin surfaces of genus g containing (C, M|C ) for 3 ≤ g ≤ 11; this completes the study of the Prym-Nikulin map initiated in [KLV].