On the leading constant in the Manin-type conjecture for Campana points

Abstract

We compare the Manin-type conjecture for Campana points recently formulated by Pieropan, Smeets, Tanimoto and Várilly-Alvarado with an alternative prediction of Browning and Van Valckenborgh in the special case of the orbifold $({\mathbb{P}}^1,D)$, where $D=\frac{1}{2}[0]+\frac{1}{2}[1]+\frac{1}{2}[\infty ]$. We find that the two predicted leading constants do not agree, and we discuss whether thin sets could explain this discrepancy. Motivated by this, we provide a counterexample to the Manin-type conjecture for Campana points, by considering orbifolds corresponding to squareful values of binary quadratic forms.

arXiv commentActa Arithmetica, to appear