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 $x^2y^3 = 1$ 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 $(\mathbb{G}_m, \tfrac{1}{2}[1])$. 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.