Symmetric products and puncturing Campana-special varieties

TL;DR

This paper provides a counterexample to existing conjectures on rational points using symmetric powers of surfaces, proposes a revised conjecture, and confirms potential density in specific cases, revealing nuanced behaviors of rational points.

Contribution

It offers a counterexample to the Arithmetic and Geometric Puncturing Conjectures and proposes a corrected conjecture inspired by Campana's ideas, advancing understanding of rational points on special varieties.

Findings

Counterexample to Puncturing Conjectures using symmetric powers

Confirmation of Campana's potential density conjecture for certain symmetric powers

Example of a surface with non-dense rational points but dense symmetric power

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.