Tame Nodal Stacky Curves

• Tame nodal stacky curves are one-dimensional Deligne–Mumford stacks with nodal coarse spaces and finite, linearly reductive stabilizers, offering a rich geometric framework.
• Their local structure is modeled by quotient and root-stack constructions, which control node behavior, smoothing families, and colliding markings.
• They play a crucial role in compactifying covers, computing Picard/Brauer invariants, and establishing categorical equivalences in homological mirror symmetry.

Tame nodal stacky curves are one-dimensional Deligne–Mumford stacks whose coarse spaces are nodal curves and whose stabilizers are finite of order prime to the characteristic; depending on the framework, the generic stabilizer may be trivial or may survive on irreducible components and then be removed by rigidification. In the literature they appear as balanced twisted curves, root stacks, quotient stacks by finite diagonalisable groups, generalized log twisted curves with colliding markings, and as curve-level manifestations of tame orders on surfaces. They serve as basic objects in compactifications of branched covers, in logarithmic and orbifold deformation theory, in Picard and Brauer computations, and in homological mirror symmetry (Deopurkar, 2015, Olsson et al., 2024, Bishop, 11 Jul 2025, Habermann, 2021).

1. Definitions and scope

A standard starting point is the balanced twisted curve of Abramovich–Vistoli. In the formulation recalled in "Covers of stacky curves and limits of plane quintics", a balanced twisted curve is a Deligne–Mumford stack $\mathcal X$ that is isomorphic to its coarse space $X$ away from finitely many points, with local charts

$$\Big[\operatorname{Spec}\big(\mathbb C[x,y]/(xy)\big)/\mu_r\Big],\qquad \zeta\cdot(x,y)=(\zeta x,\zeta^{-1}y)$$

at nodes and

$$\Big[\operatorname{Spec}\mathbb C[x]/\mu_r\Big],\qquad \zeta\cdot x=\zeta x$$

at smooth stacky points. This gives the basic notion of a nodal stacky curve: a proper, reduced, connected one-dimensional Deligne–Mumford stack whose coarse space is nodal and whose singularities are quotient singularities by finite cyclic groups. Over $\mathbb C$, tameness is automatic because all stabilizer orders are prime to the characteristic (Deopurkar, 2015).

Other papers vary the ambient definition but keep the same one-dimensional tame DM geometry. "The canonical ring of a stacky curve" works primarily with smooth proper geometrically connected Deligne–Mumford stacks of dimension $1$ with a dense open subscheme, so that stabilizers occur at finitely many points; tameness there means that stabilizer orders are not divisible by $\mathrm{char}(k)$, and in the tame case the stabilizers are cyclic $\mu_n$ (Voight et al., 2015). By contrast, "Brauer groups of tame stacky curves and $\mu_r$-gerbes over them" defines a stacky curve as a separated one-dimensional Deligne–Mumford stack of finite type with trivial generic stabilizer, allowing the coarse curve to be arbitrarily singular; its results therefore apply directly to nodal coarse curves and to stacky structure supported at nodes (Bishop, 11 Jul 2025).

A further enlargement appears in homological mirror symmetry. "Homological mirror symmetry for nodal stacky curves" allows irreducible components with nontrivial generic stabilizer $\mu_{d_i}$, so that each component is a $X$0-gerbe over a stacky projective line $X$1. In that setting the phrase “nodal stacky curve” includes chains or cycles of such gerby components glued nodally, with compatibility condition $X$2 at the nodes (Habermann, 2021). A useful organizing distinction is therefore between orbifold curves, where the generic stabilizer is trivial, and more general stacky curves, where rigidification removes generic stabilizer data.

2. Local structure, nodes, and tameness

The local geometry of tame nodal stacky curves is controlled by quotient models. For balanced twisted nodes one uses

$X$3

with the inverse characters on the two branches, and for a smoothing family one uses

$X$4

In the compactification theory of covers, the key extension statement is that a rational map $X$5 extends across the special point whenever the coarse map extends and every automorphism in the target has order dividing $X$6; this is the mechanism by which sufficiently divisible tame stabilizers force finite covers to extend across nodes (Deopurkar, 2015).

A broader toric quotient perspective is developed in "Functorial destackification of tame stacks with abelian stabilisers". There a tame nodal stacky curve is locally of the form

$X$7

with $X$8 finite diagonalisable, acting by characters on $X$9 and $$\Big[\operatorname{Spec}\big(\mathbb C[x,y]/(xy)\big)/\mu_r\Big],\qquad \zeta\cdot(x,y)=(\zeta x,\zeta^{-1}y)$$0. In that language tameness means finite inertia and linearly reductive stabilizers, and the nodal local models are one-dimensional simplicial toric quotient singularities. This is the local input for destackification and for the comparison between stacky curves and root stacks over smooth coarse curves (Bergh, 2014).

Not every paper imposes the balanced action. "Auslander orders over nodal stacky curves and partially wrapped Fukaya categories" studies balloon chains and rings in which a node is locally a quotient of $$\Big[\operatorname{Spec}\big(\mathbb C[x,y]/(xy)\big)/\mu_r\Big],\qquad \zeta\cdot(x,y)=(\zeta x,\zeta^{-1}y)$$1 by a cyclic action

$$\Big[\operatorname{Spec}\big(\mathbb C[x,y]/(xy)\big)/\mu_r\Big],\qquad \zeta\cdot(x,y)=(\zeta x,\zeta^{-1}y)$$2

so the node can be stacky without being balanced in the Abramovich–Vistoli sense. The parameters $$\Big[\operatorname{Spec}\big(\mathbb C[x,y]/(xy)\big)/\mu_r\Big],\qquad \zeta\cdot(x,y)=(\zeta x,\zeta^{-1}y)$$3 enter the resulting quiver descriptions and the mirror surface combinatorics (Lekili et al., 2017). In the gerby HMS setting, local nodes are written as

$$\Big[\operatorname{Spec}\big(\mathbb C[x,y]/(xy)\big)/\mu_r\Big],\qquad \zeta\cdot(x,y)=(\zeta x,\zeta^{-1}y)$$4

where $$\Big[\operatorname{Spec}\big(\mathbb C[x,y]/(xy)\big)/\mu_r\Big],\qquad \zeta\cdot(x,y)=(\zeta x,\zeta^{-1}y)$$5 surjects onto the branch orbifold groups on either side; this packages both node stabilizers and generic stabilizer data on adjacent components (Habermann, 2021).

Generalized log twisted curves introduce another local feature: collisions of marked points. If $$\Big[\operatorname{Spec}\big(\mathbb C[x,y]/(xy)\big)/\mu_r\Big],\qquad \zeta\cdot(x,y)=(\zeta x,\zeta^{-1}y)$$6 indexes the markings meeting at a smooth point, an admissible monoid $$\Big[\operatorname{Spec}\big(\mathbb C[x,y]/(xy)\big)/\mu_r\Big],\qquad \zeta\cdot(x,y)=(\zeta x,\zeta^{-1}y)$$7 determines the stabilizer

$$\Big[\operatorname{Spec}\big(\mathbb C[x,y]/(xy)\big)/\mu_r\Big],\qquad \zeta\cdot(x,y)=(\zeta x,\zeta^{-1}y)$$8

so colliding markings produce diagonalizable abelian stabilizers even away from nodes. The associated stack is still tame when stabilizer orders are invertible on the base (Olsson et al., 2024).

3. Root constructions, generalized log twisted curves, and moduli

A major modern framework for tame nodal stacky curves is the theory of generalized log twisted curves. Such an object is a tuple

$$\Big[\operatorname{Spec}\big(\mathbb C[x,y]/(xy)\big)/\mu_r\Big],\qquad \zeta\cdot(x,y)=(\zeta x,\zeta^{-1}y)$$9

where $$\Big[\operatorname{Spec}\mathbb C[x]/\mu_r\Big],\qquad \zeta\cdot x=\zeta x$$0 is an $$\Big[\operatorname{Spec}\mathbb C[x]/\mu_r\Big],\qquad \zeta\cdot x=\zeta x$$1-marked prestable curve, $$\Big[\operatorname{Spec}\mathbb C[x]/\mu_r\Big],\qquad \zeta\cdot x=\zeta x$$2 is a simple map of log structures controlling stackiness at nodes, and $$\Big[\operatorname{Spec}\mathbb C[x]/\mu_r\Big],\qquad \zeta\cdot x=\zeta x$$3 is an admissible monoid sheaf controlling stackiness at markings. The associated stack is

$$\Big[\operatorname{Spec}\mathbb C[x]/\mu_r\Big],\qquad \zeta\cdot x=\zeta x$$4

When the markings are distinct, admissible monoids are exactly $$\Big[\operatorname{Spec}\mathbb C[x]/\mu_r\Big],\qquad \zeta\cdot x=\zeta x$$5, and the theory recovers the usual Abramovich–Vistoli twisted curves. The moduli stack $$\Big[\operatorname{Spec}\mathbb C[x]/\mu_r\Big],\qquad \zeta\cdot x=\zeta x$$6 is a smooth algebraic stack locally of finite type, and contractions of coarse curves lift to initial contractions in the generalized log twisted category (Olsson et al., 2024).

The same root-stack philosophy governs the compactification of covers. For a family of twisted curves $$\Big[\operatorname{Spec}\mathbb C[x]/\mu_r\Big],\qquad \zeta\cdot x=\zeta x$$7 with branch divisor $$\Big[\operatorname{Spec}\mathbb C[x]/\mu_r\Big],\qquad \zeta\cdot x=\zeta x$$8 in the smooth representable locus, the stack $$\Big[\operatorname{Spec}\mathbb C[x]/\mu_r\Big],\qquad \zeta\cdot x=\zeta x$$9 parametrizes representable finite flat degree-$\mathbb C$0 covers with branch divisor $\mathbb C$1. It is a separated étale Deligne–Mumford stack of finite type, and becomes proper once the orbinode orders are sufficiently divisible. A stacky Fulton–MacPherson space $\mathbb C$2 with universal $\mathbb C$3-pointed degeneration $\mathbb C$4 supplies the boundary expansions, and the compactified moduli $\mathbb C$5 is smooth, separated, Deligne–Mumford of dimension $\mathbb C$6, proper under the same divisibility hypothesis. Boundary points are admissible covers of nodal stacky curves, and this is the mechanism used to compactify tetragonal covers and to describe the boundary of the plane quintic locus in $\mathbb C$7 (Deopurkar, 2015).

Functorial destackification gives a complementary birational picture. For a smooth tame stack with diagonalisable stabilizers and a simple normal crossings divisor, one can perform a functorial sequence of stacky blow-ups so that the final stack is a gerbe over a root stack on a smooth coarse space. Although the theorem is stated for smooth stacks, the paper explains how simplicial toric quotient singularities are treated via canonical smooth stacks; in dimension one, this turns stacky nodal geometry into root-stack data on a smooth curve with marked points, plus a possible residual gerbe (Bergh, 2014).

An important limitation is also explicit in the log-twisted framework: not every tame abelian nodal orbicurve lies in the essential image of generalized log twisted curves. The local monoid at each smooth stacky point must arise as a pushout from an admissible monoid, and examples with $\mathbb C$8-stabilizer show that this condition can fail (Olsson et al., 2024).

4. Picard groups, Brauer groups, and canonical-type invariants

For smooth tame stacky curves, the canonical divisor and canonical ring behave like orbifold versions of the classical theory. If $\mathbb C$9 is the coarse space and $1$0 is the stabilizer at a stacky point $1$1, then

$1$2

For a hyperbolic tame log stacky curve with maximal stabilizer order $1$3, the log canonical ring is generated in degree at most $1$4 with relations in degree at most $1$5; if $1$6, the bounds improve to generation in degree at most $1$7 and relations in degree at most $1$8. The paper explicitly states that analogous results are expected for nodal stacky curves after replacing $1$9 by the dualizing sheaf and using deformation theory (Voight et al., 2015).

Line bundles on stacky curves are organized by rigidification. If $\mathrm{char}(k)$0 is a gerbe banded by a finite flat generic stabilizer $\mathrm{char}(k)$1, then there is an exact sequence

$\mathrm{char}(k)$2

When the generic stabilizer is trivial and the stacky points have stabilizers $\mathrm{char}(k)$3, the smooth theory gives

$\mathrm{char}(k)$4

The paper’s root-stack identification of $\mathrm{char}(k)$5 is smooth, but the rigidification-gerbe sequence itself is general; a plausible extension to nodal coarse curves preserves the same two-step description, now with $\mathrm{char}(k)$6 replaced by the Picard group of a nodal curve and with residual gerbes at nodes contributing torsion characters (Lopez, 2023).

Brauer theory is already available for singular one-dimensional tame stacks. For a $\mathrm{char}(k)$7-gerbe $\mathrm{char}(k)$8 over a separated tame stacky curve $\mathrm{char}(k)$9 with stabilizers $\mu_n$0 at the finitely many stacky points, there is an exact sequence

$\mu_n$1

The left map is injective exactly when the gerbe is a root gerbe; the right map is surjective exactly when the inflation maps $\mu_n$2 are injective for the stabilizer extensions $\mu_n$3. If $\mu_n$4 for all $\mu_n$5, the sequence splits. For smooth stacky curves, $\mu_n$6; for singular or nodal stacky curves one gets

$\mu_n$7

so the nodes contribute explicitly through stabilizer group cohomology. The same paper proves

$\mu_n$8

for every separated tame one-dimensional algebraic stack of finite type over a field (Bishop, 11 Jul 2025).

5. Categorical, mirror-symmetric, and noncommutative interpretations

Tame nodal stacky curves admit unusually explicit derived categories. In the balloon-chain and balloon-ring constructions of Lekili–Polishchuk, the curve is built by gluing weighted projective lines $\mu_n$9 along stacky nodes. The associated Auslander order

$\mu_r$0

provides a smooth proper categorical resolution of $\mu_r$1, and $\mu_r$2 admits a full strong exceptional collection whose endomorphism algebra is a monomial quiver algebra. On the A-side, the corresponding partially wrapped Fukaya category of a punctured surface with stops has the same quiver. After localization,

$\mu_r$3

so nodal stacky curves become mirrors of punctured surfaces of arbitrary genus (Lekili et al., 2017).

Gerby components can be incorporated as well. In "Homological mirror symmetry for nodal stacky curves", each irreducible component is a $\mu_r$4-gerbe over $\mu_r$5, and the nodes are quotients $\mu_r$6. The paper constructs Auslander orders and exceptional collections in this setting and proves quasi-equivalences

$\mu_r$7

with $\mu_r$8 obtained by gluing annuli according to the stack data. For invertible two-variable singularities this yields

$\mu_r$9

so tame nodal stacky curves occur as B-models mirror to finite quotients of Milnor fibres (Habermann, 2021).

Order-theoretic approaches broaden the class further. "Non-commutative nodal curves and derived tame algebras" treats stacky cycles and chains of projective lines as special cases of noncommutative nodal curves defined by nodal orders, while "Central curves on noncommutative surfaces" shows that restricting a tame order on a surface to a central curve corresponds to taking the fiber product with the associated stacky surface. Good intersections produce hereditary orders and smooth root stacks; tangencies and passages through singular points of the discriminant produce nodal or Bass orders and singular stacky curves, including quotient-stack models for Clifford conics and skew cubics (Burban et al., 2018, Baumann et al., 2024).

6. Arithmetic and explicit families

Arithmetic applications currently center on the smooth case, but they identify the threshold phenomena that any broader theory of tame nodal stacky curves must address. For a tame stacky curve over a global field with signature $\mu_{d_i}$0, the orbifold genus is

$\mu_{d_i}$1

If $\mu_{d_i}$2 and the geometric fundamental group is finite abelian, then the Brauer–Manin obstruction is the only obstruction to strong approximation; for smooth proper integral models of genus $\mu_{d_i}$3, the elementary obstruction is the only obstruction to the integral Hasse principle. The paper computes the Brauer–Manin obstruction explicitly for genus $\mu_{d_i}$4, i.e. signature $\mu_{d_i}$5, where the universal cover is a $\mu_{d_i}$6-torsor and the relevant Brauer classes are quaternion algebras (Santens, 2022).

Bhargava and Poonen show that genus $\mu_{d_i}$7 is the first genuinely arithmetic case. They construct a proper stacky curve $\mu_{d_i}$8 of genus $\mu_{d_i}$9 over $X$00 with $X$01 and $X$02 for every prime $X$03, but $X$04; they also prove that any stacky curve of genus $X$05 over a ring of $X$06-integers of a global field satisfies the local-global principle for integral points. These results are proved for smooth stacky curves, but they isolate the low-genus regime in which tame stacky geometry becomes arithmetically rigid (Bhargava et al., 2020).

Modular stacks provide explicit tame stacky curves of a different kind. For

$X$07

the Deligne–Rapoport modular stack $X$08 has coarse space $X$09 and admits a canonical presentation

$X$10

where $X$11 is an explicit root stack over $X$12 along the divisors $X$13, $X$14, $X$15, and $X$16, and the generic stabilizer is $X$17. Local root-stack models over DVRs describe branching and stabilizer behavior in terms of ramification indices, making these modular curves a concrete supply of tame stacky curves with explicit local isotropy, rigidification, and height theory (Darda et al., 23 Feb 2026).

Taken together, these viewpoints show that tame nodal stacky curves are not a single rigid class but a family of closely related one-dimensional tame DM geometries. Their common features are a nodal coarse moduli space, finite linearly reductive stabilizers, root-stack or quotient-stack local models, and a strong compatibility between singularity theory, moduli, and categorical invariants.