The non-degeneracy invariant of Brandhorst and Shimada's families of Enriques surfaces

TL;DR

This paper investigates the non-degeneracy invariant of a broad class of Enriques surfaces, establishing lower bounds, typical values, and presenting the first example with an infinite automorphism group and a different invariant.

Contribution

It provides lower bounds for the non-degeneracy invariant, shows that most such surfaces have a generic value of 10, and presents the first example with an infinite automorphism group and a non-10 invariant.

Findings

Most $( au,ar{ au})$-generic Enriques surfaces have invariant 10.

Established lower bounds for the non-degeneracy invariant.

First example of a complex Enriques surface with infinite automorphism group and invariant not equal to 10.

Abstract

Brandhorst and Shimada described a large class of Enriques surfaces, called $(\tau,\overline{\tau})$-generic, for which they gave generators for the automorphism groups and calculated the elliptic fibrations and the smooth rational curves up to automorphisms. In the present paper, we give lower bounds for the non-degeneracy invariant of such Enriques surfaces, we show that in most cases the invariant has generic value $10$, and we present the first known example of complex Enriques surface with infinite automorphism group and non-degeneracy invariant not equal to $10$.