On the leading constant in the Manin-type conjecture for Campana points
Shute AL. On the leading constant in the Manin-type conjecture for Campana points. arXiv, 2104.14946.
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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.
Publishing Year
Date Published
2021-04-30
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arXiv
Acknowledgement
The author would like to thank Damaris Schindler and Florian Wilsch for their helpful comments on the heights and Tamagawa measures used in Section 3, together with Marta Pieropan, Sho Tanimoto and Sam Streeter for providing valuable feedback on an earlier version of this paper, and Tim Browning for many useful comments and discussions during the development of this work. The author is also grateful to the anonymous referee for providing many valuable comments and suggestions that improved the quality of the paper.
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2104.14946
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Cite this
Shute AL. On the leading constant in the Manin-type conjecture for Campana points. arXiv. doi:10.48550/arXiv.2104.14946
Shute, A. L. (n.d.). On the leading constant in the Manin-type conjecture for Campana points. arXiv. https://doi.org/10.48550/arXiv.2104.14946
Shute, Alec L. “On the Leading Constant in the Manin-Type Conjecture for Campana Points.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2104.14946.
A. L. Shute, “On the leading constant in the Manin-type conjecture for Campana points,” arXiv. .
Shute AL. On the leading constant in the Manin-type conjecture for Campana points. arXiv, 2104.14946.
Shute, Alec L. “On the Leading Constant in the Manin-Type Conjecture for Campana Points.” ArXiv, 2104.14946, doi:10.48550/arXiv.2104.14946.
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