We study the ternary relation R of collinearity between points of a smooth cubic surface X. Our aim is to understand sequences of triples (A,B,C) of finite sets with |A|=|B|=|C| whose Cartesian products have asymptotically maximal possible size of the intersection with the relation R (i.e. growing roughly as quickly as the square of the size of the sets A,B,C). This is an instance of the so-called Elekes-Szabó problem. It is well-known that such configurations can be built on planar cubic curves, hence we can find them in an intersection of X with a plane. We prove that if X is smooth and irreducible, then actually any such configuration on X concentrates on its intersection with a plane, and if X is reducible the same holds in most cases.
This is a joint work with Martin Bays and Tingxiang Zou.