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arXiv · 2412.14931

Symmetric products and puncturing Campana-special varieties

Finn Bartsch·Ariyan Javanpeykar·Aaron Levin
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Abstract

We give a counterexample to the Arithmetic Puncturing Conjecture and Geometric Puncturing Conjecture of Hassett-Tschinkel using symmetric powers of uniruled surfaces, and propose a corrected conjecture inspired by Campana's conjectures on special varieties. We confirm Campana's conjecture on potential density for symmetric powers of products of curves. As a by-product, we obtain an example of a surface without a potentially dense set of rational points, but for which some symmetric power does have a dense set of rational points, and even satisfies Corvaja-Zannier's version of the Hilbert property.

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Finn Bartsch, Ariyan Javanpeykar, Aaron Levin. 2024-12-19. Symmetric products and puncturing Campana-special varieties. https://arxiv.org/abs/2412.14931

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