On cubic surfaces with a rational line

TL;DR

This paper explores constructing smooth cubic surfaces over the rationals that contain a rational line, using degree 4 Del Pezzo surfaces and explicit Galois descent methods.

Contribution

It introduces an explicit Galois descent approach starting from diagonal degree 4 Del Pezzo surfaces to construct cubic surfaces with rational lines.

Findings

Successful construction of non-singular cubic surfaces over $bQ$ with a rational line

Development of an explicit Galois descent method for these surfaces

Potential new examples of cubic surfaces with rational lines

Abstract

We report on our project to construct non-singular cubic surfaces over $\bbQ$ with a rational line. Our method is to start with degree 4 Del Pezzo surfaces in diagonal form. For these, we develop an explicit version of Galois descent.