"Elekes-Szabó for group actions"
Abstract: Let W be an irreducible subvariety of the product V_1*V_2*V_3 of irreducible varieties over the complex numbers, such that dim(W)=dim(p_{i,j}(W))=2dim(V_i) for any projection p_{i,j} from W to V_i*V_j. We call such W a fiber-algebraic ternary relation. The Elekes-Szabó's theorem states if a fiber-algebraic ternary relation W has large intersection with arbitrarily large finite grids of the form B_1*B_2*B_3 where B_i are finite subsets of V_i, then W must be essentially the graph of an algebraic groups under the assumption that the finite sets B_i have very small intersection with all subvarieties of V_i. This assumption, called in coarse general position (cgp) is very strong in the sense that it only allows commutative algebraic groups, while the graphs of nilpotent algebraic groups are more natural examples of large intersections with finite grids. Therefore, we propose a weaker condition called in weak general position (wgp) which characterises exactly nilpotency when W is a graph of an algebraic group. In the general case, asymmetric versions of W occur naturally from compositions which lead to group actions. In this talk, I will present a group action version of the Elekes-Szabo's theorem with the cpg assumption and a characterisation of when algebraic homogeneous spaces can admit large intersections with finite sets in weak general position. These are joint works with Matin Bays and Jan Dobrowolski.
