The automorphism groups of Enriques surfaces covered by symmetric quartic surfaces

Abstract

Let $S$ be the (minimal) Enriques surface obtained from the symmetric quartic surface $({\sum }_{i<j}{x}_i{x}_j{)}^2=k{x}_1{x}_2{x}_3{x}_4$ in ${\mathbb{P}}^3$ with $k\ne 0,4,36$, 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 $S$.

arXiv comment13 pages