Mordell-Weil Problem for Cubic Surfaces, Numerical Evidence

TL;DR

This paper provides numerical evidence for the Mordell-Weil type problem on cubic surfaces and tests Manin's conjecture relating rational points and Picard rank.

Contribution

It offers the first numerical data supporting the Mordell-Weil problem for cubic surfaces and examines Manin's conjecture in this context.

Findings

Numerical evidence supports the Mordell-Weil type finiteness for certain cubic surfaces.

Data suggests a correlation between rational point distribution and Picard group rank.

Results align with predictions of Manin's conjecture for the tested surfaces.

Abstract

Let V be a plane smooth cubic curve over a finitely generated field k. The Mordell-Weil theorem for V states that there is a finite subset P \subset V(k) such that the whole V(k) can be obtained from P by drawing secants and tangents through pairs of previously constructed points and consecutively adding their new intersection points with V. In this paper we present numerical data regarding the analogous statement for cubic surfaces. For the surfaces examined, we also test Manin's conjecture relating the asymptotics of rational points of bounded height on a Fano variety with the rank of the Picard group of the surface.