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

Abstract: 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.

• The paper presents the first explicit low-dimensional example where weakly special and Campana‑special varieties diverge, answering a key question in arithmetic geometry.
• It introduces Campana stacks as a robust stack‑theoretic framework to generalize C‑pairs, resolving representability issues with orbifold data.
• The work establishes equivalences linking the density of integral points on affine varieties to Orbifold Mordell conjectures, underscoring significant arithmetic implications.

Weakly Special Varieties, Campana Stacks, and Orbifold Mordell: An Expert Synthesis

Introduction and Motivation

This work (2603.28745) rigorously analyzes and refines the gap between Campana's special varieties, the broader class of weakly special varieties, and the arithmetic of integral points, using the conceptual framework of C-pairs and Campana stacks. The authors present new explicit examples of weakly special, but not (Campana-)special, surfaces in low dimension, characterize their arithmetic implications in the context of Lang–Vojta and Orbifold Mordell conjectures, and systematically develop a stack-theoretic apparatus (Campana stacks) to encode morphisms of C-pairs beyond what root stacks provide.

Distinction between Special and Weakly Special

Let $X$ be a smooth quasi-projective variety over a field of characteristic zero. The paper provides the first explicit two-dimensional example where the notions of weakly special and Campana-special diverge: the complement $X = \mathbb{A}^2 \setminus Z(x^2y^3-1)$. The authors prove that this surface is weakly special but not special. This is significant, as prior explicit weakly special non-special varieties existed only in dimension at least three [Bogomolov–Tschinkel].

The weakly special property is defined: a variety is weakly special if, after any finite étale cover, there exist no dominant strictly rational maps to varieties of log-general type. In contrast, the Campana-special condition is stricter, disallowing dominant rational maps of general type even via resolutions, with orbifold data (via C-pairs) controlling the allowed multiplicities.

Families of Weakly Special Varieties and Their Arithmetic

A central technical achievement is the extension of the “weakly special is preserved under fibrations” criterion to quasi-projective settings with non-divisible fibres. The authors generalize previous projective results, proving that if an algebraic family $X \to S$ satisfies:

1. The base $S$ is weakly special,
2. The general fibres are weakly special,
3. No codimension-one fibre is divisible,

then the total space $X$ is weakly special. This underpins a broad range of new examples, such as affine hypersurfaces of the form $x_1^{a_1}\cdots x_n^{a_n} = 1$ with coprime exponents.

In the context of arithmetic, the paper explores the expectation that weakly special varieties over number fields are arithmetically special—i.e., have potentially dense sets of integral points. However, the authors demonstrate that this conjecture contradicts Campana's Orbifold Mordell conjecture, revealing a fundamental tension in the conjectural landscape.

Campana Stacks: Stack-Theoretic Realization of C-pairs

The study develops Campana stacks, which generalize root stacks in encoding the geometry and arithmetic of morphisms to C-pairs, especially in the general case where “at least $m$” (inf-multplicity) rather than “divisible by $m$” (gcd-multiplicity) conditions are imposed.

• The construction proceeds by building stacks over a base via a universal pushout along the moduli of effective divisors, using quotient stacks $[\mathbb{A}^1/\mathbb{G}_m]$ and their multi-root analogues.
• Campana stacks parametrizing morphisms of C-pairs resolve the non-representability issues that arise when attempting to represent subfunctors of Hom schemes directly.
• Through careful local analysis, the authors show Campana stacks and associated “Campana spaces” are regular under reasonable geometric conditions (e.g., snc divisors), and admit flat morphisms with prescribed orbifold data.

Arithmetic Equivalences and Torsor Lifting

A major arithmetic result is a set of equivalences relating the density of integral points on certain affine weakly special varieties, the arithmetic density of points on their associated Campana stacks, and the non-density of integral points on C-pairs as predicted by Orbifold Mordell. This is fully worked out, for instance, for the surface $X = \mathbb{A}^2 \setminus Z(x^2 y^3 - 1)$ and its quotient stack by a natural torus action, linking these questions to explicit forms of the abc conjecture.

Furthermore, the appendix establishes strong Chevalley–Weil type lifting theorems for torsors under group schemes, showing that potential density of integral points is preserved under passage to torsors for a wide class of algebraic groups—a pivotal point for relating the arithmetic of quotient stacks to their representability by more familiar schemes or spaces.

Generalized C-pairs and Further Generalizations

The paper goes beyond standard C-pairs by introducing generalized C-pair structures: allowing allowed multiplicities to be arbitrary finite unions of semigroups in $X = \mathbb{A}^2 \setminus Z(x^2y^3-1)$0, thus encoding more subtle and flexible “firmament” conditions (in the sense of Abramovich). Campana stacks and spaces are constructed in this generality; regularity, fibration, and arithmetic lifting results are all shown to extend compatibly.

Implications, Limitations, and Future Developments

Implications:

• The existence of explicit weakly special, non-special surfaces definitively answers the question of whether the two notions can diverge in low dimension, impacting both theoretical and practical approaches to the Lang–Vojta conjectures.
• The construction of Campana stacks provides a systematic method to encode and classify orbifold structures with arbitrary inf-multiplicity constraints, allowing generalizations of Diophantine statements and the study of arithmetic hyperbolicity to proceed in a functorial, stack-theoretic framework.
• The established equivalence between density of integral points on certain affine varieties and Orbifold Mordell-type statements indicates that progress (either positive or negative) on one side has immediate consequences for the other.

Future Directions:

• Further exploration of the geometric and arithmetic structure of Campana stacks, particularly in settings beyond characteristic zero and in the presence of more general group actions.
• Extensions of the stack-theoretic techniques to other contexts in arithmetic geometry, such as the study of fundamental groups of stacks and their relation to potential density.
• Investigation of the implications of the failure of the Weakly Special Conjecture in specific contexts, and a characterization of precisely which weakly special varieties defy arithmetic potential density.

Conclusion

This paper (2603.28745) offers a systematic, technical advance in the study of special and weakly special varieties, the definition and utility of Campana stacks, and their arithmetic ramifications. By establishing both explicit new geometric phenomena (weakly special non-special surfaces) and powerful new conceptual tools (generalized Campana stacks), the authors clarify the landscape of arithmetic hyperbolicity, provide new frameworks for Diophantine conjectures, and set the stage for further developments in the arithmetic of orbifold and stack-like structures.