Construction of Symmetric Cubic Surfaces

TL;DR

This paper classifies smooth cubic surfaces in projective 3-space with non-trivial symmetry groups, detailing their stabilizers and geometric features like Eckardt points, lines, and tritangent planes.

Contribution

It provides an explicit classification of symmetric smooth cubic surfaces and describes their automorphism groups in terms of geometric configurations.

Findings

Identified all smooth cubic surfaces with non-trivial stabilizers.

Described stabilizer groups via permutations of geometric features.

Connected group actions to geometric configurations like Eckardt points.

Abstract

We consider the action of the group $\mathrm{PGL}_4(K)$ on the smooth cubic surfaces of $\mathbb{P}^3_K$ ($K$ an algebraically closed field of characteristic zero). We classify, in an explicit way, all the smooth cubic surfaces with non trivial stabilizer, the corresponding stabilizers and obtain a geometric description of each group in terms of permutations of the Eckardt points, of the $27$ lines or of the $45$ tritangent planes.