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

Qile Chen; Brian Lehmann; Sho Tanimoto

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arXiv · 2605.19898

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

Qile Chen·Brian Lehmann·Sho Tanimoto

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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.

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BibTeX RIS

Qile Chen, Brian Lehmann, Sho Tanimoto. 2026-05-19. Manin's conjecture for semi-integral curves and $\mathbb A^1$-connectedness. https://arxiv.org/abs/2605.19898

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