Passing from marked objects to an unmarked moduli space identifies objects related by symmetry. The resulting coarse point records the orbit, but it can forget how much symmetry the representative carries. Generic objects may have trivial automorphism group while special objects have nontrivial stabilisers and different local quotient behaviour.
This guide develops that missing layer. It explains orbit types, stabiliser strata, orbifold points and why a classification that preserves only coarse points can be correct as a set while incomplete as a moduli statement.
The Gold Standard is an automorphism-sensitive claim. It states which group acts, which stabilisers occur, how local models change across strata, whether stack or orbifold data is retained and exactly what the chosen invariant can reconstruct.
A practical orientation
A stabiliser consists of symmetries that fix a chosen representative. Points with conjugate stabilisers often form an orbit-type stratum. The quotient can be smooth on one stratum and singular or orbifold-like where symmetry jumps.
The worked example uses the reflection action of C2 on the complex line. The coarse coordinate u=z² looks simple, but the origin retains a C2 automorphism in the quotient stack.
- Define the symmetry problem — for readers moving from marked to unmarked moduli.
- Read stabiliser strata — for local quotient and orbit-type analysis.
- Follow the C2 example — for explicit orbits, invariants and isotropy.
- Test classification claims — for automorphism-sensitive reconstruction.
- Transfer to moduli — for curves, bundles and configuration spaces.
Chapter contents
Open a route that matches the decision in front of you, or read straight through. Each route returns to the same evidence chain.
Start with the operating question
Build the evidence chain
Build the orbit-type or stabiliser stratification
1. Name the objects, equivalences and acting group
Begin with a marked parameter space X and a group G that changes markings or presentations. The unmarked coarse space seeks orbits X/G. A point x has stabiliser Gx={g in G | g·x=x}. This subgroup records automorphisms visible in the chosen presentation.
State whether stabilisers are considered literally or up to conjugacy, and whether the action is proper, finite, free or has fixed points. Without these hypotheses, words such as orbifold, quotient manifold and stratum can mean different things.
- Marked object and allowed markings.
- Acting group and action law.
- Equivalence represented by an orbit.
- Stabiliser and conjugacy convention.
- Coarse quotient, homotopy quotient or stack target.
- Local hypotheses used for slices or orbifold charts.
2. Separate coarse orbit data from automorphism data
The coarse quotient sends every orbit to one point. It can preserve useful invariants and topology while forgetting the isotropy attached to that point. A quotient stack or groupoid remembers objects together with their automorphisms and arrows.
This distinction is not a complaint that coarse spaces are wrong. They answer a different question. A coarse classification may identify isomorphism classes, while deformation counts, local multiplicities, integration formulas or functorial reconstruction require stabiliser information.
| Object | What it keeps | What it can forget |
|---|---|---|
| Coarse quotient X/G | Orbit points and quotient topology | Stabiliser groups and arrows |
| Homotopy quotient EG×G X | Action data up to homotopy | Some strict geometric structure |
| Quotient stack [X/G] | Objects, families and automorphisms | Requires stack-aware language and equivalence |
3. Build the orbit-type or stabiliser stratification
Group points whose stabilisers are conjugate. Under suitable proper-action hypotheses, these orbit types form locally closed pieces and satisfy frontier relations: more symmetric types often occur in the closure of less symmetric strata.
Do not stratify only by stabiliser order when nonconjugate subgroups have the same size. Record the representation of the stabiliser on a normal slice when local quotient geometry matters. Equal groups can act differently on nearby directions.
4. Recognise orbifold points and symmetry jumps
For a finite effective local action, a point with nontrivial stabiliser has an orbifold chart modelled on a neighbourhood modulo that finite group. The underlying coarse space can be topologically smooth even though the orbifold point carries nontrivial isotropy.
Symmetry jumps are special, not necessarily pathological. They often mark configurations with extra automorphisms. The local model, deformation directions and invariant functions should be checked before labeling the coarse point singular or regular.
| Stabiliser | Local action | Coarse appearance | Retained stack data |
|---|---|---|---|
| Trivial | Free near point | Ordinary local quotient | No isotropy |
| Finite reflection/rotation | Nonfree isolated symmetry | May be smooth or cone-like | Finite automorphism group |
| Positive-dimensional | Continuous fixed symmetry | Often more singular | Higher-dimensional automorphisms |
| Same order, different representation | Different normal action | Potentially different local type | Representation-sensitive isotropy |
5. Use slices to connect global orbits with local geometry
A slice is a local transverse model to the orbit. Near x, the quotient is governed by the stabiliser acting on the slice rather than by the full group acting on all of X. This is why stabiliser representations control local strata.
The existence and strength of slice models depend on hypotheses. Do not transfer a finite-group orbifold conclusion to an arbitrary nonproper action. State whether the result is topological, smooth, complex analytic or algebraic.
6. Worked example: C2 acting by z ↦ −z
Let C2={1,σ} act on the complex line C by σ(z)=−z. For z≠0, the stabiliser is trivial because z=−z would force z=0. At the origin, the entire C2 fixes the point. Thus there is a generic trivial-stabiliser stratum and one special C2 stratum.
The invariant coordinate u=z² identifies z and −z, so the coarse quotient is again a complex line. As a coarse space, the origin does not look singular. In the quotient stack [C/C2], however, the point over u=0 has automorphism group C2. A coarse coordinate alone cannot reconstruct that isotropy.
| Representative | Orbit | Stabiliser | Coarse coordinate | Automorphism-sensitive conclusion |
|---|---|---|---|---|
| z≠0 | {z,−z} | Trivial | u=z²≠0 | Generic unstacky point |
| z=0 | {0} | C2 | u=0 | Orbifold/stacky point |
| Nearby pair | Two marked choices | Trivial | One coarse point | Marking forgotten |
| Origin deformation | Normal sign action | C2 representation | Smooth coarse coordinate | Isotropy retained only beyond coarse space |
7. Add a changed-context moduli example
Consider an ordered configuration whose labels are later forgotten. A generic configuration may have no nontrivial permutation preserving it, while a symmetric configuration can be fixed by a permutation. Both become points of the unlabelled coarse moduli space, but their automorphism groups differ.
The correct stabiliser depends on the full object and allowed equivalences, not visual symmetry alone. For curves, bundles or maps, automorphisms must respect the structure in the moduli problem. Record markings, weights, sections and other decorations before computing the group.
8. Test automorphism-sensitive classification statements
A proposed classification should say whether it classifies coarse isomorphism classes, orbit types, objects with automorphism groups, or families. Test injectivity on coarse points, stabiliser preservation and local deformation compatibility separately.
Two invariants can agree on every coarse point yet assign different isotropy or gluing. Conversely, an automorphism-aware functor may be essentially surjective but fail to be fully faithful on morphisms. A complete claim needs the level of equivalence appropriate to the target.
- Coarse-point bijection.
- Stabiliser-group preservation up to isomorphism.
- Compatibility with conjugacy and local representations.
- Full faithfulness on automorphisms and arrows.
- Behaviour in families and under base change.
- Declared limits outside the tested strata.
9. Use invariants with a stated reconstruction boundary
Invariant functions often describe a coarse quotient. Add stabiliser type, slice representation or groupoid data when reconstruction requires more. Do not claim that a coordinate ring or numerical signature recovers an orbifold structure unless a theorem supplies that bridge.
For a restricted subcategory, coarse invariants plus known stabiliser data may be complete. State the restriction. A classification can be excellent on the trivial-stabiliser stratum and deliberately incomplete at symmetry jumps.
10. Know the limits of the stratified picture
Orbit-type stratifications require hypotheses and may have multiple connected components for one stabiliser type. Closure relations can be subtle, and coarse singularities need not correspond one-to-one with nontrivial stabilisers.
Stacky or orbifold information also does not solve every moduli problem. Nonseparatedness, continuous automorphisms, obstruction theories and derived structure may require richer tools. Preserve the boundary between the worked finite-action model and those settings.
11. Misconceptions, practical questions and transfer
Common mistakes are equating a smooth coarse point with trivial automorphism, stratifying only by group order, forgetting the stabiliser representation and treating all quotient stacks as orbifolds. Each mistake collapses a different layer of the action.
Transfer the method to a moduli space of Higgs bundles or weighted curves by identifying the actual automorphism group, orbit-type pieces and local model. Do not copy the C2 action unless the problem genuinely has that symmetry.
Can a coarse quotient be smooth at a point with nontrivial stabiliser?
Yes. The C2 action z↦−z has coarse coordinate u=z², so the underlying quotient is a complex line while the origin retains C2 isotropy in the quotient stack.
Is stabiliser order enough to determine the local quotient?
No. Nonconjugate subgroups or different representations of the same group can produce different local geometry. The slice action matters.
12. Changed-context evidence check and authoritative sources
A strong check identifies the acting group, computes generic and special stabilisers, tests conjugacy, describes a local slice action and states what the coarse invariant forgets. If the classification claims families or equivalence of moduli problems, verify arrows and automorphisms rather than only points.
Yue Fan’s orbit-type stratification of the moduli space of Higgs bundles proves a Whitney stratification with complex symplectic strata in that setting. Abdelgadir, Chan, Okawa and Ueda’s compact moduli of orbifold projective curves relates orbifold projective curves to weighted pointed stable curves using automorphism groups of stacky points. These papers support the specialist examples; the elementary C2 computation above is self-contained.
Continue through the Gold Standard ecosystem
Return to the Topology subject root, the Gold Standard of Education parent, or the How X Works hub.
Immediate predecessor: marked versus unmarked moduli and homotopy quotients · Earlier guide: moduli-component separation and automorphism actions · Topology root
These routes are grouped by learning purpose: prerequisite model, immediate predecessor, and the wider transfer framework. They are not substitutes for the evidence and limits stated in this guide.