Combinatorial cubic surfaces and reconstruction theorems

TL;DR

This paper presents a method to reconstruct the defining field and equation of a projective cubic surface solely from combinatorial data about its rational points, using collinearity and coplanarity relations.

Contribution

It introduces a novel combinatorial approach to reconstruct cubic surfaces, advancing understanding without explicit use of model theoretic language.

Findings

Successfully reconstructs the defining field and equation from combinatorial data.

Provides conditions under which the reconstruction is possible.

Lays groundwork for addressing the Mordell–Weil problem via combinatorial methods.

Abstract

This note contains a solution to the following problem: reconstruct the definition field and the equation of a projective cubic surface, using only combinatorial information about the set of its rational points. This information is encoded in two relations: collinearity and coplanarity of certain subsets of points. We solve this problem, assuming mild ``general position'' properties. This study is motivated by an attempt to address the Mordell--Weil problem for cubic surfaces using essentially model theoretic methods. However, the language of model theory is not used explicitly.