On moduli spaces of polarized Enriques surfaces

We prove that, for any g >= 2, the etale double cover rho(g) : epsilon(g) -> (epsilon) over cap (g) from the moduli space epsilon(g) of complex polarized genus g Enriques surfaces to the moduli space (epsilon) over cap (g) of numerically polarized genus g Enriques surfaces is disconnected precisely over irreducible components of (epsilon) over cap (g) parametrizing 2-divisible classes, answering a question of Gritsenko and Hulek [13]. We characterize all irreducible components of epsilon(g) in terms of a new invariant of line bundles on Enriques surfaces that generalizes the phi-invariant introduced by Cossec [8]. In particular, we get a one-to-one correspondence between the irreducible components of epsilon(g) and 11-tuples of integers satisfying particular conditions. This makes it possible, in principle, to list all irreducible components of epsilon(g) for each g >= 2. (C) 2020 The Author. Published by Elsevier Masson SAS.