Yifan Jing
The Ohio State University
Title
A projective Szemerédi–Trotter theorem
Abstract
The Szemerédi–Trotter theorem asserts that the solutions of an algebraic curve from a given set in a field of characteristic 0 exhibit a power-saving improvement over the Schwartz–Zippel bound, provided that the curve satisfies a K_{s,t}-free condition and the set is close to being a cube with equal side lengths. We establish a group-action version of the Szemerédi–Trotter theorem over any field, extending an earlier result by Bourgain. As an application, we obtain quantitative bounds on the number of collinear triples on reducible cubic surfaces in \mathbb{P}^3(k), where k = \mathbb{F}_p^n or \mathbb{C}, which can be interpreted as a quantitative Elekes–Szabó-type theorem without a general position assumption. (based on joint work with Tingxiang Zou)
