Abstract

For an Enriques surface S, the non-degeneracy invariant $\text{nd}(S)$ retains information on the elliptic fibrations of S and its polarizations. In the current paper, we introduce a combinatorial version of the non-degeneracy invariant which depends on S together with a configuration of smooth rational curves, and gives a lower bound for $\text{nd}(S)$. We provide a SageMath code that computes this combinatorial invariant and we apply it in several examples. First we identify a new family of nodal Enriques surfaces satisfying $\text{nd}(S)=10$ which are not general and with infinite automorphism group. We obtain lower bounds on $\text{nd}(S)$ for the Enriques surfaces with eight disjoint smooth rational curves studied by Mendes Lopes–Pardini. Finally, we recover Dolgachev and Kondō’s computation of the non-degeneracy invariant of the Enriques surfaces with finite automorphism group and provide additional information on the geometry of their elliptic fibrations.

摘要

对于 Enriques 曲面S，非退化不变量$\text{nd}(\mathrm{小号})$保留有关S的椭圆纤维化及其极化的信息。在本文中，我们介绍了非退化不变量的组合版本，它依赖于S以及光滑有理曲线的配置，并给出了$\text{nd}(\mathrm{小号})$. 我们提供了一个 SageMath 代码来计算这个组合不变量，并在几个例子中应用它。首先，我们确定了一个新的节点 Enriques 曲面家族，满足$\text{nd}(\mathrm{小号})=10$它们不是一般的并且具有无限的自同构群。我们得到下界$\text{nd}(\mathrm{小号})$对于 Mendes Lopes-Pardini 研究的具有 8 条不相交的光滑有理曲线的 Enriques 曲面。最后，我们恢复了 Dolgachev 和 Kondō 对具有有限自同构群的 Enriques 曲面的非退化不变量的计算，并提供了有关其椭圆纤维几何的额外信息。