Congruence subgroups and Enriques surface automorphisms

TL;DR

This paper provides conceptual proofs for bounds on the automorphism group of Enriques surfaces, extending previous computational results and applying over various algebraically closed fields.

Contribution

It offers new conceptual proofs for bounds on automorphism groups of Enriques surfaces, improving understanding of their structure without relying on computational methods.

Findings

Established a lower bound close to the known upper bound for the automorphism group's image.

Results apply over any algebraically closed field under specific conditions on nodal curves.

Extended previous work by providing conceptual rather than computational proofs.

Abstract

We give conceptual proofs of some results on the automorphism group of an Enriques surface X, for which only computational proofs have been available. Namely, there is an obvious upper bound on the image of Aut(X) in the isometry group of X's numerical lattice, and we establish a lower bound for the image that is quite close to this upper bound. These results apply over any algebraically closed field, provided that X lacks nodal curves, or that all its nodal curves are (numerically) congruent to each other mod 2. In this generality these results were originally proven by Looijenga and Cossec-Dolgachev, developing earlier work of Coble.