Weakly special varieties, Campana stacks, and Remarks on Orbifold Mordell

We construct the first weakly special surfaces that are not Campana-special, including the complement of the plane curve $x^2{y}^3=1$ in ${\mathbb{A}}^2$. We prove that the set of ${\mathcal{O}}_{K,S}$-integral points on this surface is non-dense for every number field $K$ and finite set $S$ of finite places of $K$ if and only if Campana’s Orbifold Mordell conjecture holds for $({\mathbb{G}}_m,\frac{1}{2}[1])$. This basic example carries a natural ${\mathbb{G}}_m$-action, and the quotient stack is an Artin stack parametrizing points on a C-pair. This leads to the introduction of ``Campana stacks’', which encode morphisms of C-pairs in a manner analogous to the role of root stacks for integral points satisfying prescribed divisibility conditions.