On moduli spaces of polarized Enriques surfaces
Journal article, Peer reviewed
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Original versionJournal des Mathématiques Pures et Appliquées. 2020, 144, 106-136. 10.1016/j.matpur.2020.10.003
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 . 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 . 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.