On the Borisov-Nuer conjecture and the image of the Enriques-to-K3 map

Abstract

We discuss the Borisov-Nuer conjecture in connection with the canonical maps from the moduli spaces ${\mathcal{M}}_{En,h}^a$of polarized Enriques surfaces with fixed polarization type $h$ to the moduli space ${\mathcal{F}}_g$ of polarized $K3$ surfaces of genus $g$ with $g=h^2+1$, and we exhibit a naturally defined locus ${\Sigma }_g\subset {\mathcal{F}}_g$. One direct consequence of the Borisov-Nuer conjecture is that ${\Sigma }_g$ would be contained in a particular Noether-Lefschetz divisor in ${\mathcal{F}}_g$, which we call the Borisov-Nuer divisor and we denote by ${\mathcal{B}\mathcal{N}}_g$. In this short note, we prove that ${\Sigma }_g\cap {\mathcal{B}\mathcal{N}}_g$ is non-empty whenever $(g-1)$ is divisible by $4$. To this end, we construct polarized Enriques surfaces $(Y,H_Y)$, with $H_Y^2$ divisible by $4$, which verify the conjecture. In particular, the conjecture holds also for any element ${\mathcal{M}}_{En,h}^a$, if $h^2$ is divisible by $4$ and $h$ is the same type of polarization.

11 pages