On the number of Mordell-Weil generators for cubic surfaces

TL;DR

This paper investigates the minimal number of K-rational points needed to generate all points on a smooth cubic surface using secant and tangent lines, proving a special case of the Mordell-Weil conjecture.

Contribution

It introduces an abelian group associated with cubic surfaces and proves that if the surface contains two skew lines over K, then only one generator is needed.

Findings

If the surface contains two skew lines over K, then r(S,K) = 1.

Constructs a family of cubic surfaces with unbounded r(S,K).

Introduces the group H_S(K) to analyze the secant and tangent process.

Abstract

Let S be a smooth cubic surface over a field K. It is well-known that new K-rational points may be obtained from old ones by secant and tangent constructions. A Mordell-Weil generating set is a subset B of S(K) of minimal cardinality which generates S(K) via successive secant and tangent constructions. Let r(S,K) be the cardinality of such a Mordell-Weil generating set. Manin posed what is known as the Mordell-Weil problem for cubic surfaces: if K is finitely generated over its prime subfield then r(S,K) is finite. In this paper, we prove a special case of this conjecture. Namely, if S contains two skew lines both defined over K then r(S,K) = 1. One of the difficulties in studying the secant and tangent process on cubic surfaces is that it does not lead to an associative binary operation as in the case of elliptic curves. As a partial remedy we introduce an abelian group H_S(K) associated to a cubic surface S/K, naturally generated by the K-rational points, which retains much information about the secant and tangent process. In particular, r(S, K) is large as soon as the minimal number of generators of H_S(K) is large. In situations where weak approximation holds, H_S has nice local-to-global properties. We use these to construct a family of smooth cubic surfaces over the rationals such that r(S,K) is unbounded in this family. This is the cubic surface analogue of the unboundedness of ranks conjecture for elliptic curves.