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.

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关于鲍里索夫-努尔猜想和 Enriques-to-K3 地图的图像

我们讨论了鲍里索夫-努尔猜想，结合从模空间 $\mathcal M_{En,h}^a$ 的极化 Enriques 表面具有固定的极化类型 $h$ 到模空间 $\mathcal F_g$ 的典型映射来讨论鲍里索夫-努尔猜想。用 $g=h^2+1$ 极化 $g$ 属的 $K3$ 表面，我们展示了一个自然定义的轨迹 $\Sigma_g\subset\mathcal F_g$。Borisov-Nuer 猜想的一个直接结果是 $\Sigma_g$ 将包含在 $\mathcal F_g$ 中的特定 Noether-Lefschetz 除数中，我们将其称为 Borisov-Nuer 除数，并用 $\mathcal{BN} 表示_g$。在这个简短的注释中，我们证明只要 $(g-1)$ 可以被 $4$ 整除，$\Sigma_g\cap\mathcal{BN}_g$ 就是非空的。为此，我们构建了极化的恩里克斯面 $(Y, H_Y)$，其中 $H_Y^2$ 可以被 $4$ 整除，从而验证了猜想。特别是，