Moduli of curves on toric varieties and their stable cohomology

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.

arXiv admin note: text overlap with arXiv:2207.02496