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 ℳ𝑎𝐸𝑛,ℎ of 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 𝐻2𝑌 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′,𝐻′𝑌) contained in the same moduli.