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

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.