A group-action Szemer\'edi-Trotter theorem and applications to orchard problems in all characteristics

TL;DR

This paper extends the Szemerédi-Trotter theorem to group actions over any field, providing new bounds on collinear triples on cubic surfaces, with implications for combinatorial geometry in various characteristics.

Contribution

It introduces a group-action version of the Szemerédi-Trotter theorem applicable over all fields, generalizing previous results and improving bounds on geometric configurations.

Findings

Established a group-action Szemerédi-Trotter theorem over any field.

Derived quantitative bounds on collinear triples on cubic surfaces.

Extended results to both finite fields and complex numbers.

Abstract

We establish a group-action version of the Szemer\'edi-Trotter theorem over any field, extending Bourgain's result for the group $\mathrm{SL}_2(k)$. As an Elekes-Szab\'o-type application, we obtain quantitative bounds on the number of collinear triples on reducible cubic surfaces in $\mathbb{P}^3(k)$, where $k = \mathbb{F}_{q}$ and $k = \mathbb{C}$, thereby improving a recent result by Bays, Dobrowolski, and the second author.