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We discuss the Borisov-Nuer conjecture in connection with the canonical mapsfrom the moduli spaces $\mathcal M_{En,h}^a$of polarized Enriques surfaces withfixed 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 definedlocus $\Sigma_g\subset\mathcal F_g$. One direct consequence of the Borisov-Nuerconjecture is that $\Sigma_g$ would be contained in a particularNoether-Lefschetz divisor in $\mathcal F_g$, which we call the Borisov-Nuerdivisor 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 conjectureholds also for any element $\mathcal M_{En,h}^a$, if $h^2$ is divisible by $4$and $h$ is the same type of polarization.

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