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arXiv · 2202.01775

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

Riccardo Moschetti·Franco Rota·Luca Schaffler
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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.

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Riccardo Moschetti, Franco Rota, Luca Schaffler. 2022-02-03. A computational view on the non-degeneracy invariant for Enriques surfaces. https://doi.org/10.1080/10586458.2022.2113576

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