On moduli spaces of polarized Enriques surfaces

We prove that, for any $g\ge 2$, the 'etale double cover ${\rho }_g:{\mathcal{E}}_g\to {\widehat{\mathcal{E}}}_g$ from the moduli space ${\mathcal{E}}_g$ of complex polarized genus $g$ Enriques surfaces to the moduli space ${\widehat{\mathcal{E}}}_g$ of numerically polarized genus $g$ Enriques surfaces is disconnected precisely over irreducible components of ${\widehat{\mathcal{E}}}_g$ parametrizing $2$-divisible classes, answering a question of Gritsenko and Hulek. We characterize all irreducible components of ${\mathcal{E}}_g$ in terms of a new invariant of line bundles on Enriques surfaces that generalizes the $\varphi$-invariant introduced by Cossec. In particular, we get a one-to-one correspondence between the irreducible components of ${\mathcal{E}}_g$ and $11$-tuples of integers satisfying particular conditions. This makes it possible, in principle, to list all irreducible components of ${\mathcal{E}}_g$ for each $g\ge 2$.