More counterexamples to the Arithmetic Puncturing Problem
Finn Bartsch
TL;DR
The paper tackles the arithmetic puncturing problem for Campana-special varieties by constructing explicit low-dimensional counterexamples with terminal or canonical singularities. It uses quotient constructions to produce affine and projective surfaces and threefolds that are Campana-special and exhibit a dense set of integral points, dense entire curves, vanishing Kobayashi pseudometric, and geometric specialness, yet whose regular loci fail to retain any of these properties. A key result is that some examples satisfy weak approximation, while their regular loci do not satisfy strong approximation off any finite set of places, signaling a negative answer to puncturing questions in the singular setting. These findings illuminate how singularities disrupt the expected equivalences between arithmetic, analytic, and geometric notions and refine our understanding of Campana-special varieties under codimension-two puncturing.
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.