Invariant Smooth Quartic Surfaces by all Finite Primitive Groups of ${\mathrm{PGL}}_4(\mathbb{C})$

Abstract

For each finite primitive subgroup $G$ of ${\mathrm{PGL}}_4(\mathbb{C})$, we find all the smooth $G$-invariant quartic surfaces. We also find all the faithful representations in ${\mathrm{PGL}}_4(\mathbb{C})$ of the smooth quartic $G$-invariant surfaces by the groups: ${\mathfrak{A}}_5,{\mathfrak{S}}_5,\mathrm{PS}{\mathrm{L}}_2\left({\mathbb{F}}_7\right),{\mathfrak{A}}_6,{\mathbb{Z}}_2^4⋊{\mathbb{Z}}_5$ and ${\mathbb{Z}}_2^4⋊D_{10}$. The primitive representation of these groups are precisely the subgroups of ${\mathrm{PGL}}_4(\mathbb{C})$ for which ${\mathbb{P}}^3$ is not $G$-super rigid. As a byproduct, we show that the smooth quartic surface with the biggest group of projective automorphism is given by $\{x_0^4+x_1^4+x_2^4+x_3^4+12x_0{x}_1{x}_2{x}_3=0\}$ (unique up to projective equivalence).

22 pages, comments are welcome