A group-action Szemerédi-Trotter theorem and applications to orchard problems in all characteristics

We establish a group-action version of the Szemerédi-Trotter theorem over any field, extending Bourgain’s result for the group ${\mathrm{S}\mathrm{L}}_2(k)$. As an Elekes-Szabó-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.