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  ,ℎ of polarized Enriques surfaces with fixed ℎ ∈ ⊕ 8 (−1) to the moduli space  of polarized 3 surfaces of genus with = ℎ 2 + 1, and we exhibit a naturally defined locus Σ ⊂  . One direct consequence of the Borisov-Nuer conjecture is that Σ would be contained in a particular Noether-Lefschetz divisor in  , which we call the Borisov-Nuer divisor and we denote by  . In this short note, we prove that Σ ∩  is non-empty whenever ( − 1) is divisible by 4. To this end, we construct polarized Enriques surfaces ( , ) , with 2 divisible by 4, which verify the conjecture. In particular, when we consider the moduli space of (numerically) polarized Enriques surfaces which contains such ( , ), the conjecture also holds for any other polarized Enriques surface ( ′ , ′ ) contained in the same moduli. K E Y W O R D SBorisov-Nuer conjecture, Enriques surface, Jacobian Kummer surface, numerically polarized Enriques surface M S C ( 2 0 1 0 ) 14D20, 14J28