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

TL;DR

This paper classifies all smooth quartic surfaces in complex projective 3-space that are invariant under finite primitive groups of projective transformations, identifying their automorphism groups and providing explicit examples.

Contribution

It systematically determines all smooth G-invariant quartic surfaces for finite primitive subgroups of PGL(4,C), including explicit group actions and a unique maximally symmetric example.

Findings

Classified all smooth G-invariant quartic surfaces for specified groups.

Identified the primitive representations of these groups in PGL(4,C).

Found the quartic surface with the largest automorphism group.

Abstract

For each finite primitive subgroup $G$ of $\operatorname{PGL}_4(\mathbb{C})$, we find all the smooth $G$-invariant quartic surfaces. We also find all the faithful representations in $\operatorname{PGL}_4(\mathbb{C})$ of the smooth quartic $G$-invariant surfaces by the groups: $\mathfrak{A}_5,\mathfrak{S}_5, \operatorname{PSL_2(\mathbb{F}_7)},\mathfrak{A}_6,\mathbb{Z}_2^4\rtimes\mathbb{Z}_5$ and $\mathbb{Z}_2^4\rtimes D_{10}$. The primitive representation of these groups are precisely the subgroups of $\operatorname{PGL}_4(\mathbb{C})$ 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 $\{ x_0^4 + x_1^4 + x_2^4 + x_3^4 + 12 x_0 x_1 x_2 x_3= 0\}$ (unique up to projective equivalence).