Manin's conjecture for semi-integral curves and $\mathbb A^1$-connectedness

TL;DR

This paper investigates log Manin's conjecture for integral points, establishing results for Campana rational curves and $ ext{A}^1$-curves on split toric varieties, using geometric methods and Cox rings.

Contribution

It proves log Manin's conjecture for specific classes of curves on toric varieties, providing a geometric explanation for the leading constant in the conjecture.

Findings

Proved log Manin's conjecture for Campana rational curves.

Established results for $ ext{A}^1$-curves on split toric varieties.

Provided a geometric interpretation of the leading constant.

Abstract

We explore log Manin's conjecture for integral points and its connections to $\mathbb A^1$-connectedness. We prove log Manin's conjecture for Campana rational curves and for $\mathbb A^1$-curves on split toric varieties. Our arguments combine the Cox ring description of the moduli space of rational curves with Batyrev's heuristic-type counting arguments. As our proofs are geometric in nature, they give a geometric explanation of the mysterious leading constant for Campana points proposed by Chow--Loughran--Takloo-Bighash--Tanimoto.