The automorphism groups of Enriques surfaces covered by symmetric quartic surfaces

TL;DR

This paper investigates the automorphism groups of certain Enriques surfaces derived from symmetric quartic surfaces, revealing their structure as semi-direct products and classifying elliptic pencils on these surfaces.

Contribution

It explicitly determines the automorphism group structure of Enriques surfaces covered by symmetric quartic surfaces and classifies their elliptic pencils.

Findings

Automorphism group is a semi-direct product of a free product of four involutions and S4.

Exactly 29 elliptic pencils up to the action of the involutions.

Provides explicit description of automorphism groups for these surfaces.

Abstract

Let $S$ be the (minimal) Enriques surface obtained from the symmetric quartic surface $(\sum_{i<j}x_ix_j)^2=kx_1x_2x_3x_4$ in $\mathbb{P}^3$ with $k\neq 0,4,36$, by taking quotient of the Cremona action $(x_i) \mapsto (1/x_i)$. The automorphism group of $S$ is a semi-direct product of a free product $\mathcal{F}$ of four involutions and the symmetric group $\mathfrak{S}_4$. Up to action of $\mathcal{F}$, there are exactly $29$ elliptic pencils on $S$.