More counterexamples to the Arithmetic Puncturing Problem

TL;DR

This paper constructs specific threefolds and surfaces with particular singularities that exhibit complex arithmetic and geometric properties, providing new counterexamples to the Arithmetic Puncturing Problem and refining previous results.

Contribution

It introduces new examples of singular varieties with dense integral points and entire curves, advancing understanding of the Arithmetic Puncturing Problem and related conjectures.

Findings

Constructed threefolds with terminal singularities exhibiting dense integral points.

Provided surfaces with canonical singularities that are geometrically special but lack certain properties.

Showed some examples satisfy the weak approximation property.

Abstract

We construct examples of threefolds with terminal singularities (resp. surfaces with canonical singularities) which are special in the sense of Campana, have a potentially dense set of integral points, admit a dense entire curve, have vanishing Kobayashi pseudometric, and are geometrically special in the sense of Javanpeykar-Rousseau but whose regular locus fails to have any of these properties. This improves on earlier work by Cadorel-Campana-Rousseau and joint work by the author with Javanpeykar-Levin, where such fourfolds with canonical singularities were constructed, and gives refined answers to questions due to Hassett-Tschinkel and Kamenova-Lehn. Lastly, we show that some of our examples satisfy the weak approximation property and briefly discuss a question on puncturing varieties satisfying strong approximation raised by Wittenberg.