Geometric Manin's conjecture in characteristic $p$

TL;DR

This paper explores a version of Geometric Manin's conjecture in characteristic p, connecting it to broader conjectures over global fields, and provides a survey of related topics in algebraic geometry.

Contribution

It introduces a characteristic p analogue of Geometric Manin's conjecture and discusses its relation to existing conjectures over global fields.

Findings

Proposes a new version of the conjecture in characteristic p

Establishes connections with the Batyrev--Manin--Peyre--Tschinkel conjecture

Provides a comprehensive survey of the topic

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.