Combinatorial cubic surfaces and Mordell-Weil

 The geometric version of weak Mordell-Weil problem for cubic surfaces over number fields,- finite generation of the set of rational points,- still remains unsolved problem. At the HIM workshop "Diophantine Equations", Spring 2009, I formulated the program of a new approach to this problem, with model theoretic flavor. I will discuss this approach and report on a recent progress (Yu.Manin, arXiv:1001.0223): the notion of a combinatorial cubic surface and a Reconstruction Theorem, producing the definition field and the surface itself from the relations of collinearity and coplanarity on the set of rational points.