Cubic surfaces with a Galois invariant pair of Steiner trihedra

TL;DR

This paper introduces a method to construct smooth cubic surfaces over the rationals that possess a Galois invariant pair of Steiner trihedra, utilizing explicit Galois descent on generalized Cayley-Salmon forms.

Contribution

It provides a new explicit Galois descent technique for cubic surfaces with specific Galois-invariant configurations, extending classical forms.

Findings

Constructed examples of cubic surfaces with Galois invariant Steiner trihedra.

Developed an explicit Galois descent method for generalized Cayley-Salmon forms.

Enhanced understanding of Galois actions on cubic surface configurations.

Abstract

We present a method to construct non-singular cubic surfaces over $\bbQ$ with a Galois invariant pair of Steiner trihedra. We start with cubic surfaces in a form generalizing that of A. Cayley and G. Salmon. For these, we develop an explicit version of Galois descent.