Moduli of curves on toric varieties and their stable cohomology

TL;DR

This paper proves the stabilization of the cohomology of moduli spaces of morphisms from curves to toric varieties and applies this to resolve the Batyrev-Manin conjecture over global function fields.

Contribution

It establishes cohomology stabilization for these moduli spaces and provides a resolution of the Batyrev-Manin conjecture in almost all characteristics.

Findings

Cohomology of moduli spaces stabilizes as degree varies.

Resolution of the Batyrev-Manin conjecture for toric varieties over global function fields.

Applicable in all but finitely many characteristics.

Abstract

We prove that the cohomology of the moduli space of morphisms of a fixed finite degree from a smooth projective curve $C$ of genus $g$ to a complete simplicial toric variety $\mathbb{P}(\Sigma)$, denoted by the rational polyhedral fan $\Sigma$, stabilizes. As an arithmetic consequence we obtain a resolution of the Batyrev-Manin conjecture for toric varieties over global function fields in all but finitely many characteristics.