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

We discuss the Borisov–Nuer conjecture in connection with the canonical maps from the moduli spaces MEn,haof polarized Enriques surfaces with fixed h∈U⊕E8(−1) to the moduli space Fg of polarized K3 surfaces of genus g with g=h2+1, and we exhibit a naturally defined locus Σg⊂Fg. One direct consequence of the Borisov–Nuer conjecture is that Σg would be contained in a particular Noether–Lefschetz divisor in Fg, which we call the Borisov–Nuer divisor and we denote by BNg. In this short note, we prove that Σg∩BNg is non‐empty whenever (g−1) is divisible by 4. To this end, we construct polarized Enriques surfaces (Y,HY), with HY2 divisible by 4, which verify the conjecture. In particular, when we consider the moduli space of (numerically) polarized Enriques surfaces which contains such (Y,HY), the conjecture also holds for any other polarized Enriques surface (Y′,HY′) contained in the same moduli.