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

TL;DR

This paper compares two predictions for the leading constant in the Manin-type conjecture for Campana points, finds discrepancies, and provides a counterexample involving orbifolds and squareful values of quadratic forms.

Contribution

It identifies a disagreement between two conjectural constants and offers a counterexample to the conjecture using orbifolds related to quadratic forms.

Findings

The two conjectures predict different leading constants for the orbifold case.

Thin sets may not fully explain the discrepancy.

Counterexample shows the conjecture does not always hold.

Abstract

We compare the Manin-type conjecture for Campana points recently formulated by Pieropan, Smeets, Tanimoto and V\'{a}rilly-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.