For an Enriques surface S, the non-degeneracy invariant 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 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 nd(S) = 10 which are not general and with infinite automorphism group. We obtain lower bounds on nd(S) for the Enriques surfaces with eight disjoint smooth rational curves studied by Mendes Lopes-Pardini. Finally, we recover Dolgachev and Kondo's computation of the non-degeneracy invariant of the Enriques surfaceswith finite automorphism group and provide additional information on the geometry of their elliptic fibrations.

Moschetti, R., Rota, F., Schaffler, L. (2022). A Computational View on the Non-degeneracy Invariant for Enriques Surfaces. EXPERIMENTAL MATHEMATICS, 1-22 [10.1080/10586458.2022.2113576].

A Computational View on the Non-degeneracy Invariant for Enriques Surfaces

Luca Schaffler
2022-01-01

Abstract

For an Enriques surface S, the non-degeneracy invariant 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 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 nd(S) = 10 which are not general and with infinite automorphism group. We obtain lower bounds on nd(S) for the Enriques surfaces with eight disjoint smooth rational curves studied by Mendes Lopes-Pardini. Finally, we recover Dolgachev and Kondo's computation of the non-degeneracy invariant of the Enriques surfaceswith finite automorphism group and provide additional information on the geometry of their elliptic fibrations.
Moschetti, R., Rota, F., Schaffler, L. (2022). A Computational View on the Non-degeneracy Invariant for Enriques Surfaces. EXPERIMENTAL MATHEMATICS, 1-22 [10.1080/10586458.2022.2113576].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11590/423987
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