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

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.