More counterexamples to the Arithmetic Puncturing Problem

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.