A computational view on the non-degeneracy invariant for Enriques surfaces

TL;DR

This paper introduces a combinatorial approach to compute the non-degeneracy invariant of Enriques surfaces, providing new bounds and classifications, and implements it in SageMath for practical calculations.

Contribution

It presents a novel combinatorial invariant for Enriques surfaces, along with a SageMath implementation, and applies it to identify new families and analyze automorphism groups.

Findings

Identified a new family of nodal Enriques surfaces with non-degeneracy invariant 10

Provided lower bounds for surfaces with eight disjoint rational curves

Reproduced and extended previous computations of the invariant for surfaces with finite automorphism groups

Abstract

For an Enriques surface $S$, the non-degeneracy invariant $\mathrm{nd}(S)$ 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 $\mathrm{nd}(S)$. 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 $\mathrm{nd}(S)=10$ which are not general and with infinite automorphism group. We obtain lower bounds on $\mathrm{nd}(S)$ for the Enriques surfaces with eight disjoint smooth rational curves studied by Mendes Lopes-Pardini. Finally, we recover Dolgachev and Kond\=o'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.