Symmetric products and puncturing Campana-special varieties

2024/12/19 by Finn Bartsch, Bartsch, Finn, Ariyan Javanpeykar +3 · 3 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2412.14931

openalex publication_date 2024/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Hyperbolicity and fundamental groups of complex quasi-projective varieties (III): applications
• Hilbert properties of varieties
• Fibration theorems for varieties with the weak Hilbert property

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