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

## TL;DR

This paper explores the Borisov-Nuer conjecture relating to the moduli spaces of polarized Enriques and K3 surfaces, proving non-emptiness of a specific intersection under certain divisibility conditions on the polarization.

## Contribution

It demonstrates that the intersection of the locus  with the Borisov-Nuer divisor is non-empty when the genus-related parameter is divisible by 4, constructing explicit polarized Enriques surfaces.

## Key findings

- The intersection   is non-empty for (g-1) divisible by 4.
- Constructed polarized Enriques surfaces verify the Borisov-Nuer conjecture under divisibility conditions.
- The conjecture holds for polarizations with square divisible by 4.

## 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 $\mathcal{BN}_g$. In this short note, we prove that $\Sigma_g\cap\mathcal{BN}_g$ 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.