On an Enriques surface associated with a quartic Hessian surface

TL;DR

This paper investigates the automorphism group of a specific Enriques surface linked to a quartic Hessian surface, providing explicit descriptions of its structure, fundamental domain, elliptic fibrations, and rational double points.

Contribution

It offers a finite presentation of the automorphism group of such Enriques surfaces and explicitly describes the fundamental domain and elliptic fibrations.

Findings

Finite presentation of the automorphism group obtained.

Explicit fundamental domain of the automorphism group action described.

List of elliptic fibrations and rational double points provided.

Abstract

Let Y be a complex Enriques surface whose universal cover X is birational to a general quartic Hessian surface. Using the result on the automorphism group of X due to Dolgachev and Keum, we obtain a finite presentation of the automorphism group of Y. A fundamental domain of the action of the automorphism group on the nef cone of Y is described explicitly. The list of elliptic fibrations on Y and the list of combinations of rational double points that can appear on a surface birational to Y are presented. As an application, a set of generators of the automorphism group of the generic Enriques surface is calculated explicitly.