Combinatorial cubic surfaces and reconstruction theorems

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.