Tame Stacky Node

• Tame stacky node is a local quotient model of a nodal curve by a finite cyclic group under conditions ensuring tame (prime-to-characteristic) stabilization.
• It is classified into twisted and doubly-twisted nodes with explicit numerical parameters that govern local inertia, Picard groups, and Brauer contributions.
• The framework unifies quotient, logarithmic, and root-stack approaches to describe orbifold structures and deformation behavior in modular curve settings.

Searching arXiv for the cited papers to ground the article in current records. A tame stacky node is the local stack structure at a node of a tame Deligne–Mumford stacky curve such that, after passing to a strict henselization or étale neighborhood, the node is modeled by a quotient of the ordinary nodal curve $\operatorname{Spec}\mathbbm k[x,y]/(xy)$ by a finite stabilizer group. In the split-node setting, the local model is $[\operatorname{Spec} R/G]$, where $R$ is the strict henselization of the local ring of $\mathbbm k[x,y]/(xy)$ at the origin and $G$ is either cyclic or branch-swapping (Bishop et al., 25 Sep 2025). In logarithmic formulations, the same phenomenon is encoded by a simple morphism of log structures and appears as the cyclic orbifold chart $$\left[\operatorname{Spec}(k[u,v]/(uv))/\mu_r\right]$$ with weights $(1,-1)$ (Olsson et al., 2024). The tame condition means that the relevant stabilizer orders are prime to the residue characteristics, so the local geometry is governed by cyclic inertia, root stacks, rigidifications, and representable ramification rather than by wild higher-ramification data (Darda et al., 23 Feb 2026).

1. Local notion and ambient hypotheses

In the narrow sense developed for nodal stacky curves, a nodal stacky curve is a proper, geometrically integral, one-dimensional Deligne–Mumford stack $\mathcal C$ over $\mathbbm k$ with trivial generic stabilizer, such that at each closed geometric point the completed strict henselian local ring is either $\bar{\mathbbm k}[[x]]$ or $[\operatorname{Spec} R/G]$0. The analysis of tame stacky nodes in this setting assumes that all nodes are split, meaning that the node itself and its preimages in the normalization are $[\operatorname{Spec} R/G]$1-points (Bishop et al., 25 Sep 2025).

A broader logarithmic framework treats node stackiness as one part of a generalized log twisted curve

$[\operatorname{Spec} R/G]$2

where the simple inclusion $[\operatorname{Spec} R/G]$3 produces stackiness at nodes and the admissible sheaf

$[\operatorname{Spec} R/G]$4

produces stackiness at markings. The associated stack is

$[\operatorname{Spec} R/G]$5

with coarse space $[\operatorname{Spec} R/G]$6 and abelian stabilizers supported at nodes and markings (Olsson et al., 2024).

The tameness hypothesis is decisive in each of these formulations. For tame Deligne–Mumford stacks and tame stacky curves, geometric stabilizer orders are prime to the characteristic of the base field, which is precisely what permits cyclic quotient models, root-stack descriptions, and tractable Picard- and Brauer-theoretic calculations (Bishop, 11 Jul 2025). In the modular-curve setting, this same tame behavior is built into the cyclotomic Deligne–Mumford formalism: the relevant local phenomena are finite cyclic inertia at stacky points together with ramification indices in representable maps, and the paper is explicit that these local stacky phenomena are not wild (Darda et al., 23 Feb 2026).

2. Classification of local models

The classification theorem for tame stacky nodes in the split-node case is completely explicit. After strict henselization,

$[\operatorname{Spec} R/G]$7

and there are exactly two kinds of local stabilizer structure (Bishop et al., 25 Sep 2025).

Type	Stabilizer	Local description
Twisted node	$[\operatorname{Spec} R/G]$8	$[\operatorname{Spec} R/G]$9, $R$0
Doubly-twisted node	$R$1	$R$2 swaps branches, with $R$3

A twisted node is thus a cyclic quotient

$R$4

with $R$5. Many authors reserve “twisted node” for the balanced case

$R$6

but in this classification $R$7 is allowed to be any unit modulo $R$8 (Bishop et al., 25 Sep 2025). The paper introduces the numerical parameters

$R$9

which control both local inertia behavior and the Picard contribution of the node.

A doubly-twisted node has local structure

$\mathbbm k[x,y]/(xy)$0

where the quotient $\mathbbm k[x,y]/(xy)$1 exchanges the two branches and the cyclic subgroup still acts by

$\mathbbm k[x,y]/(xy)$2

The branch-swapping involution forces

$\mathbbm k[x,y]/(xy)$3

The local group is not determined by $\mathbbm k[x,y]/(xy)$4 alone: if $\mathbbm k[x,y]/(xy)$5 lifts the nontrivial element of $\mathbbm k[x,y]/(xy)$6, then

$\mathbbm k[x,y]/(xy)$7

for a generator $\mathbbm k[x,y]/(xy)$8 of $\mathbbm k[x,y]/(xy)$9, with $G$0. The parameter $G$1 is well-defined modulo $G$2, lies in the kernel of multiplication by $G$3, determines the extension class

$G$4

and the extension is split iff

$G$5

The notation for this group is $G$6, and every admissible triple $G$7 occurs on some nodal stacky curve (Bishop et al., 25 Sep 2025).

An arithmetic dichotomy governs these local models: $G$8 This distinction later controls when the node contributes nontrivially to the Brauer group (Bishop et al., 25 Sep 2025).

3. Logarithmic, root-stack, and ramification descriptions

In logarithmic language, a nodal coarse curve is locally given by the smoothing equation

$G$9

with canonical log chart

$$\left[\operatorname{Spec}(k[u,v]/(uv))/\mu_r\right]$$0

The associated tame stacky node has the orbifold chart

$$\left[\operatorname{Spec}(k[u,v]/(uv))/\mu_r\right]$$1

with action

$$\left[\operatorname{Spec}(k[u,v]/(uv))/\mu_r\right]$$2

This is the standard cyclic tame node model in the generalized log twisted-curve construction (Olsson et al., 2024).

The same paper makes the deformation theory concrete under contraction. If a node with stabilizer $$\left[\operatorname{Spec}(k[u,v]/(uv))/\mu_r\right]$$3 is contracted to one with smaller stabilizer $$\left[\operatorname{Spec}(k[u,v]/(uv))/\mu_r\right]$$4, where $$\left[\operatorname{Spec}(k[u,v]/(uv))/\mu_r\right]$$5, the local map is

$$\left[\operatorname{Spec}(k[u,v]/(uv))/\mu_r\right]$$6

with

$$\left[\operatorname{Spec}(k[u,v]/(uv))/\mu_r\right]$$7

This gives an explicit local rule for how tame stacky node structure changes under contraction (Olsson et al., 2024).

Root stacks furnish the corresponding smooth-point and branch-point models. For tame stacky curves with trivial generic stabilizer, the local form near a stacky point is

$$\left[\operatorname{Spec}(k[u,v]/(uv))/\mu_r\right]$$8

and globally such curves are identified with iterated root stacks along stacky points (Lopez, 2023). This does not itself give a nodal classification, but it isolates the same cyclic Kummer mechanism that underlies tame node charts.

In the modular-curve setting, rigidification and root-stack structure are carried out explicitly. For the Deligne–Rapoport stack $$\left[\operatorname{Spec}(k[u,v]/(uv))/\mu_r\right]$$9, the core structural statement is

$(1,-1)$0

obtained from

$(1,-1)$1

by base change along $(1,-1)$2 (Darda et al., 23 Feb 2026). The local model after rigidification is a root stack over a smooth formal disc,

$(1,-1)$3

with closed-point automorphism group $(1,-1)$4; the paper also writes

$(1,-1)$5

and, for the $(1,-1)$6-gerbe structure,

$(1,-1)$7

This is a branch-point model rather than a nodal one, but it uses the same tame cyclic and root-stack technology (Darda et al., 23 Feb 2026).

4. Normalization, coarsening, Picard groups, and Brauer groups

The coarsening and normalization of a tame stacky node separate the singular quotient from the residual cyclic stabilizer structure. If $(1,-1)$8 is the coarse moduli map and $(1,-1)$9 is the partial normalization, then in the doubly-twisted case the étale local structure of the normalization is

$\mathcal C$0

Thus the normalization is smooth with cyclic stabilizer $\mathcal C$1, and its composition with $\mathcal C$2 is the coarse moduli map of $\mathcal C$3 (Bishop et al., 25 Sep 2025).

The local node type is visible in the Picard group. For a twisted node,

$\mathcal C$4

and ultimately

$\mathcal C$5

In the balanced case $\mathcal C$6,

$\mathcal C$7

For a doubly-twisted node,

$\mathcal C$8

The $\mathcal C$9 factor is the branch-swap contribution, and the root order is governed by $\mathbbm k$0 (Bishop et al., 25 Sep 2025).

The Brauer-theoretic behavior is sharper. A twisted node contributes nothing because

$\mathbbm k$1

A doubly-twisted node can contribute a $\mathbbm k$2 only in the split case and only when

$\mathbbm k$3

Otherwise the local Brauer contribution is zero (Bishop et al., 25 Sep 2025).

A more global cohomological formulation treats any singular stacky curve pointwise through stabilizer cohomology. For a separated tame Deligne–Mumford stack $\mathbbm k$4 with coarse moduli map $\mathbbm k$5, the local stalk formula is

$\mathbbm k$6

and for a stacky curve

$\mathbbm k$7

The paper notes explicitly that the $\mathbbm k$8 contributing in this way must be singular points, since smooth stacky points in a tame stacky curve have only cyclic stabilizers (Bishop, 11 Jul 2025). This provides a cohomological criterion for distinguishing ordinary tame stacky points from genuinely singular stacky nodes.

5. Arithmetic realization on modular curves

The modular stack $\mathbbm k$9 supplies an arithmetic setting in which tame local stacky structure controls global height geometry. For

$\bar{\mathbbm k}[[x]]$0

these are precisely the values for which the coarse moduli space of $\bar{\mathbbm k}[[x]]$1 is isomorphic to $\bar{\mathbbm k}[[x]]$2, and the stacky Batyrev–Manin conjecture is proved for the naive height when $\bar{\mathbbm k}[[x]]$3 (Darda et al., 23 Feb 2026).

For these stacks, the generic stabilizer is always $\bar{\mathbbm k}[[x]]$4, coming from the involution $\bar{\mathbbm k}[[x]]$5 on elliptic curves. The only points with larger stabilizer lie above the special $\bar{\mathbbm k}[[x]]$6-values $\bar{\mathbbm k}[[x]]$7 and $\bar{\mathbbm k}[[x]]$8, where the automorphism groups become $\bar{\mathbbm k}[[x]]$9 and $[\operatorname{Spec} R/G]$00, respectively. Their counts are encoded by

$[\operatorname{Spec} R/G]$01

and

$[\operatorname{Spec} R/G]$02

The ordinary canonical degree is then

$[\operatorname{Spec} R/G]$03

which is exactly the contribution of the tame stacky points with stabilizers $[\operatorname{Spec} R/G]$04 and $[\operatorname{Spec} R/G]$05 (Darda et al., 23 Feb 2026).

The orbifold structure is recorded by

$[\operatorname{Spec} R/G]$06

There is always the untwisted sector $[\operatorname{Spec} R/G]$07, and there is the ubiquitous sector $[\operatorname{Spec} R/G]$08 of age $[\operatorname{Spec} R/G]$09 coming from the generic $[\operatorname{Spec} R/G]$10-stabilizer. Over points with automorphism group $[\operatorname{Spec} R/G]$11 or $[\operatorname{Spec} R/G]$12, there are additional twisted sectors: $[\operatorname{Spec} R/G]$13 and

$[\operatorname{Spec} R/G]$14

These sectors enter the orbifold Néron–Severi space $[\operatorname{Spec} R/G]$15 and determine the orbifold corrections to the canonical class and height class (Darda et al., 23 Feb 2026).

The local ramification formulas are equally explicit. For a representable map of root stacks over DVRs,

$[\operatorname{Spec} R/G]$16

one has

$[\operatorname{Spec} R/G]$17

Specialized formulas for the $[\operatorname{Spec} R/G]$18 and $[\operatorname{Spec} R/G]$19 cases are

$[\operatorname{Spec} R/G]$20

with

$[\operatorname{Spec} R/G]$21

and

$[\operatorname{Spec} R/G]$22

with

$[\operatorname{Spec} R/G]$23

These formulas determine which sector a stacky point contributes to and thereby feed into the orbifold $[\operatorname{Spec} R/G]$24- and $[\operatorname{Spec} R/G]$25-invariants for the naive height (Darda et al., 23 Feb 2026).

6. Tame versus wild behavior and related frameworks

The tame classification depends on the fact that cyclic stabilizer order suffices to describe the local quotient geometry. In characteristic $[\operatorname{Spec} R/G]$26, this fails for cyclic stabilizers of order $[\operatorname{Spec} R/G]$27. The local root-stack model

$[\operatorname{Spec} R/G]$28

must then be replaced by an Artin–Schreier root stack

$[\operatorname{Spec} R/G]$29

and the local quotient is governed by an Artin–Schreier equation

$[\operatorname{Spec} R/G]$30

together with the ramification jump $[\operatorname{Spec} R/G]$31 (Kobin, 2019). The paper’s central point is that, unlike the tame case, the order $[\operatorname{Spec} R/G]$32 of the stabilizer does not determine the local structure: one must record higher ramification data. A plausible implication is that the clean two-type classification of tame stacky nodes cannot extend verbatim to the wild cyclic case.

This contrast clarifies the role of root stacks in the tame theory. For tame stacky curves with trivial generic stabilizer, the local quotient form

$[\operatorname{Spec} R/G]$33

underlies Picard computations and rigidification arguments (Lopez, 2023). That paper does not give a separate theory of nodal stacky curves, but it isolates the same cyclic Kummer mechanism that reappears at tame nodes. This suggests that tame stacky nodes should be viewed as the nodal analogue of tame stacky points, with the ordinary node $[\operatorname{Spec} R/G]$34 replacing the smooth local parameter.

At the level of birational and toroidal geometry, smooth tame stacks with diagonalisable stabilisers admit a functorial destackification algorithm by sequences of ordinary blow-ups and root stacks, ending with a smooth coarse space and a residual gerbe over a root stack of a simple normal crossings divisor (Bergh, 2014). This is not a classification of nodal stacky curves, but it places tame local quotient singularities into a broader framework in which stackiness is systematically reduced to root-stack data. In that sense, the tame stacky node sits at the intersection of three parallel formalisms: quotient classification, logarithmic twisted curves, and root-stack rigidification.