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

Abstract

A. For an Enriques surface S, the non-degeneracy invariant nd(S) retains information on the elliptic brations 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 con guration of smooth rational curves, and gives a lower bound for nd(S). We provide a SageMath code that computes such 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 in nite 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 Kondō's computation of the non-degeneracy invariant of the Enriques surfaces with nite automorphism group and provide additional information on the geometry of their elliptic brations.

Enriques surfaces of non-degeneracy 3

“…Consequently, one can list all genus one fibrations on X, and compute the exact values of min nd(X) and max nd(X) (cf. [10,21]). For the sake of completeness, these values are displayed in 4.2.…”