Affine Stacky Curves

• Affine stacky curves are smooth, one-dimensional Deligne–Mumford stacks that generalize affine curves by incorporating local orbifold and ramification data.
• They are constructed via cyclic (tame) and Artin–Schreier (wild) root stacks, and their classification hinges on detailed ramification jumps and stack inertia configurations.
• The study of these curves advances anabelian and arithmetic geometry by linking étale fundamental groups and rigidification sequences to moduli and number theory.

An affine stacky curve is a one-dimensional, smooth, irreducible Deligne–Mumford stack that is generically a scheme but features finitely many closed points with nontrivial finite inertia. Such objects generalize affine algebraic curves to naturally encode local orbifold and wild ramification data. Their structure, classification, and invariants substantially differ between characteristic zero (where all inertia is tame) and positive characteristic (where wild ramification emerges). The study of affine stacky curves integrates local root stack constructions, Artin–Schreier theory, and rigidified moduli spaces, and it enables increasingly fine-grained anabelian and arithmetic investigations.

1. Foundational Definitions: Affine Stacky Curves and Inertia

An affine stacky curve $X$ over a field $K$ is a smooth, separated, connected Deligne–Mumford stack of finite type, generically a scheme, admitting a global presentation as a quotient $[{\rm Spec}\,A / G]$, where $A$ is a smooth one-dimensional $K$-algebra and $G$ is a finite group scheme acting on ${\rm Spec}\,A$ and acting generically trivially (Kobin, 2019). The geometry of $X$ is dictated by its coarse moduli space $\overline{X}$ (a smooth affine curve) and by the collection of finite stack inertia groups $I_{X,\bar{x}}$ attached to closed geometric points $K$0; the generic inertia $K$1 is the inertia group at the generic point and is the maximal finite normal subgroup of the étale fundamental group $K$2 (Collas et al., 4 May 2026).

Every affine stacky curve admits a rigidification $K$3, the maximal stack quotient where only non-generic inertia remains. This leads to the exact rigidification sequence: $K$4 Combined with the coarsification sequence, which relates to the full coarse moduli scheme, these structures enable detection of the stacky curve’s global and local inertia configurations.

2. Local and Global Constructions: Root Stacks and Artin–Schreier Theory

In characteristic zero (or when the order of inertia is invertible in the base field), affine stacky curves locally arise as quotient stacks by stabilizer groups $K$5 attached along divisors; these are constructed via cyclic (tame) root stacks,

$K$6

where $K$7 is a line bundle on the coarse curve $K$8, $K$9 is a global section, and $[{\rm Spec}\,A / G]$0 acts via $[{\rm Spec}\,A / G]$1, $[{\rm Spec}\,A / G]$2 (Kobin, 2019).

In positive characteristic $[{\rm Spec}\,A / G]$3, tame root stacks fail: the map $[{\rm Spec}\,A / G]$4 is inseparable for $[{\rm Spec}\,A / G]$5, and the tame stack is not Deligne–Mumford. Instead, wild ramification of order $[{\rm Spec}\,A / G]$6 is parametrized through Artin–Schreier root stacks, given by the local model: $[{\rm Spec}\,A / G]$7 with the group action $[{\rm Spec}\,A / G]$8 for $[{\rm Spec}\,A / G]$9. This construction precisely encodes higher ramification data via the polynomial $A$0 and its valuations at zeros and poles, capturing wild ramification jumps undetectable by tame root stacks. These local Artin–Schreier covers glue globally through additive cocycles $A$1, mirroring Kummer theory but with the Artin–Schreier additive structure (Kobin, 2019).

3. Classification: Ramification Jumps, Stacky Points, and Isomorphism Types

The isomorphism class of an affine stacky curve with $A$2 stabilizers at finitely many points is determined by the locations and ramification jumps of its stacky points. For a point $A$3 with inertia $A$4, the ramification filtration on $A$5 has a unique jump $A$6: writing the extension locally as $A$7, and $A$8, the jump is $A$9. The global class is then dictated by the coarse curve and the multiset of positive integers $K$0 at the stacky points, a classification much finer than that afforded by stabilizer order alone (Kobin, 2019).

In characteristic zero, the analogous classification data is simpler: the order and location of each stacky point suffice, as all ramification is tame.

4. Canonical Divisors, Riemann–Roch, and Wild Ramification

The canonical divisor on an affine stacky curve incorporates both ordinary and stacky contributions. In the Artin–Schreier setting, wild ramification ensures that the canonical divisor $K$1 is a $K$2-divisor, typically expressed as: $K$3 where $K$4 is the set of stacky points, $K$5 are terms in the ramification filtration, and multiplicities depend on the particular jump profile (Kobin, 2019). The degree of $K$6 can reflect fractional contributions, unlike the tame case. Cohomological invariants such as $K$7, for $K$8 a divisor, are computed via the “floor” of the $K$9-divisor, restoring classical results in the tame case but with refined behavior for wild ramification.

Explicit calculations (e.g., for $G$0, $G$1 on $G$2) exhibit stacky curves whose geometric genus, as determined by these invariants, differs from the coarse genus. For instance, a stacky curve above $G$3 with two order-3 stacky points of jump $G$4 has “stacky genus” one, aligning its space of global differentials with that of an elliptic curve (Kobin, 2019).

5. Anabelian Reconstruction: Fundamental Groups and Stepwise Detection

Affine Deligne–Mumford curves are completely classified, up to isomorphism over finitely generated fields, by their étale fundamental groups modulo the fifth term of their derived series ($G$5-step solvable quotients) (Collas et al., 4 May 2026). Gross features such as hyperbolicity and affineness are detectable at the $G$6-step level:

• Hyperbolicity: Defined via the orbifold Euler characteristic,

$G$7

hyperbolicity is characterized by its negativity and can be read off from $G$8 (Collas et al., 4 May 2026).

• Affineness: Affine curves correspond to those with at least one “cusp” ($G$9), equivalently when ${\rm Spec}\,A$0 is affine. This property is likewise deduced from the solvable ${\rm Spec}\,A$1-step quotient.
• Full isomorphism class: The ${\rm Spec}\,A$2-step quotient ${\rm Spec}\,A$3 recovers both stack inertia and the extension/gerbe class over the rigidification. The subsets of torsion cycles in ${\rm Spec}\,A$4 encode local periods, while higher steps kill nontrivial extension classes to canonically specify the stack data.

This anabelian theory generalizes the program of Tamagawa, Mochizuki, and Oda–Tamagawa, showing that the functor from affine stacky curves to their ${\rm Spec}\,A$5-step fundamental group quotients is fully faithful and essentially surjective in the hyperbolic, affine case (Collas et al., 4 May 2026).

6. Examples: Weighted Projective Lines and Cyclic Inertia

Examples highlight the algorithm. For an affine stacky curve structure on ${\rm Spec}\,A$6 with two orbifold points of order ${\rm Spec}\,A$7 and a cusp, the fundamental group’s abelianization is ${\rm Spec}\,A$8; affineness is detected at the ${\rm Spec}\,A$9-step level; the gerbe class vanishes at the $X$0-step level, uniquely specifying the stack structure (Collas et al., 4 May 2026).

Weighted projective lines $X$1 provide non-affine, orbifold examples. Their fundamental group is the profinite completion of the triangle group $X$2. All stacky data—the periods $X$3, the absence of cusps, and the extension class—are extracted iteratively from the derived series quotients of the fundamental group.

7. Application and Future Directions

The rigidification and coarsification sequences of affine stacky curves provide a complete functorial package of their stack-theoretic and arithmetic structure for applications in anabelian geometry, moduli of curves, and the study of wild ramification. The $X$4-step anabelian correspondence yields a weak Grothendieck conjecture for stacks: isomorphism classes over a number field correspond to isomorphism classes of the $X$5-step fundamental group quotients as Galois representations (Collas et al., 4 May 2026). Extension of these results to higher-dimensional moduli stacks and log–Fano orbifolds is anticipated, with foundational implications for the arithmetic and moduli theory of stacks and configurations spaces.

The interplay between local Artin–Schreier data and global stack cohomology, combined with solvable-step anabelian results, positions affine stacky curves as a robust framework for both wild ramification theory in positive characteristic and fine arithmetic invariants over number fields.