Geometric Manin's conjecture in characteristic $p$

Abstract

Geometric Manin's conjecture for complex Fano varieties describes the structure of the moduli space of curves. We propose a version of this conjecture in characteristic $p$ and describe its connection to the Batyrev--Manin--Peyre--Tschinkel conjecture over global fields. This is a survey paper written for a volume of the Summer Research Institute in Algebraic Geometry held at Colorado State University in 2025.

the exposition has been improved, 38 pages, to appear in 2025 Summer Research Institute in Algebraic Geometry proceedings volume