Generators for Cubic Surfaces with two Skew Lines over Finite Fields

TL;DR

This paper extends previous results on generating rational points on smooth cubic surfaces over finite fields, showing that under certain conditions, all points can be generated from a small set, with minimal field size requirements.

Contribution

It proves that for fields with at least 4 elements, a single point can generate all rational points on the surface if it contains a skew pair of lines, broadening earlier results.

Findings

For fields with at least 4 elements, S(K) can be generated from one point.

Siksek's result for fields with at least 13 elements is extended to smaller fields.

Slightly milder results are obtained for fields with 2 or 3 elements.

Abstract

Let S be a smooth cubic surface defined over a field K. As observed by Segre and Manin, there is a secant and tangent process on S that generates new K-rational points from old. It is natural to ask for the size of a minimal generating set for S(K). In a recent paper, for fields K with at least 13 elements, Siksek showed that if S contains a skew pair of K-lines then S(K) can be generated from one point. In this paper we prove the corresponding version of this result for fields K having at least 4 elements, and slightly milder results for #K=2 or 3.