The Elekes-Szabó problem for cubic surfaces
The Elekes-Szabó problem asks when a complex variety V⊆ ∏i=13Wi has unexpected large intersections with Cartesian products of finite subsets Xi⊆ Wi for 1≤ i≤ 3. Under the assumption that Xi’s are in general position, Elekes and Szabó proved that one can always find commutative algebraic groups in this scenario. We explored the case when Wi’s are a fixed cubic surface S in ℙ3(ℂ) and V is the collinearity relation with the assumption that Xi does not concentrate on any one-dimensional subvarieties of S, which substantially weakens the general position assumption. We proved that when S is a smooth quadric surface union a plane, then one cannot find such Xi’s. When S is an irreducible smooth cubic surface, then Xi’s would contain a union of translates of arithmetic progressions on the family of planar cubic curves of S. But the existence of such Xi’s is still open. This is a work-in-progress joint with Martin Bays and Jan Dobrowolski.
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