On moduli spaces of polarized Enriques surfaces

Abstract

We prove that, for any g≥2, the étale double cover ρg:Eg→̂Eg from the moduli space Egof complex polarized genus g Enriques surfaces to the moduli space ̂Egof numerically polarized genus g Enriques surfaces is disconnected precisely over irreducible components of ̂Eg parametrizing 2-divisible classes, answering a question of Gritsenko and Hulek [13]. We characterize all irreducible components of Egin terms of a new invariant of line bundles on Enriques surfaces that generalizes the φ-invariant introduced by Cossec [8]. In particular, we get a one-to-one correspondence between the irreducible components of Egand 11-tuples of integers satisfying particular conditions. This makes it possible, in principle, to list all irreducible components of Eg for each g≥2.