On moduli spaces of polarized Enriques surfaces

We prove that, for any $g \geq 2$, 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 $\phi$-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 $g \geq 2$.