Canonical reduction of stabilizers for Artin stacks with good moduli spaces
aa r X i v : . [ m a t h . AG ] O c t CANONICAL REDUCTION OF STABILIZERS FOR ARTIN STACKSWITH GOOD MODULI SPACES
DAN EDIDIN AND DAVID RYDH
Abstract.
We present a complete generalization of Kirwan’s partial desingulariza-tion theorem on quotients of smooth varieties. Precisely, we prove that if X is anirreducible Artin stack with stable good moduli space X π → X , then there is a canon-ical sequence of birational morphisms of Artin stacks X n → X n − → . . . → X = X with the following properties: (1) the maximum dimension of a stabilizer of a pointof X k +1 is strictly smaller than the maximum dimension of a stabilizer of X k andthe final stack X n has constant stabilizer dimension; (2) the morphisms X k +1 → X k induce proper and birational morphisms of good moduli spaces X k +1 → X k . If inaddition the stack X is smooth, then each of the intermediate stacks X k is smoothand the final stack X n is a gerbe over a tame stack. In this case the algebraic space X n has tame quotient singularities and is a partial desingularization of the good modulispace X .When X is smooth our result can be combined with D. Bergh’s recent destack-ification theorem for tame stacks to obtain a full desingularization of the algebraicspace X . Contents
1. Introduction 22. Stable good moduli spaces and statement of the main theorem 53. Saturated Proj and saturated blowups 84. Reichstein transforms and saturated blowups 145. Equivariant Reichstein transforms and fixed points 166. The proof of Theorem 2.11 in the smooth case 197. Corollaries of Theorem 2.11 in the smooth case 238. The proof of Theorem 2.11 in the singular case 25Appendix A. Gerbes and good moduli spaces 29Appendix B. Fixed loci of Artin stacks 31References 34
Date : Oct 1, 2018.2010
Mathematics Subject Classification.
Primary 14D23; Secondary 14E15, 14L24.
Key words and phrases.
Good moduli spaces, Kirwan’s partial desingularization, Reichstein trans-forms, geometric invariant theory.The first author was partially supported by the Simons collaboration grant 315460 while preparingthis article. The second author was partially supported by the Swedish Research Council, grants2011-5599 and 2015-05554. Introduction
Consider the action of a reductive group G on a smooth projective variety X . Forany ample G -linearized line bundle on X there is a corresponding projective geometricinvariant theory (GIT) quotient X//G . If X s = X ss , then X//G has finite quotientsingularities. However, if X s = X ss , then the singularities of X//G can be quite bad.In a classic paper, Kirwan [Kir85] used a careful analysis of stable and unstable pointson blowups to prove that if X s = ∅ , then there is a sequence of blowups along smoothcenters X n → X n − → . . . → X = X with the following properties: (1) The finalblowup X n is a smooth projective G -variety with X sn = X ssn . (2) The map of GITquotients X n //G → X//G is proper and birational. Since X n //G has only finite quotientsingularities, we may view it as a partial resolution of the very singular quotient X//G .Kirwan’s result can be expressed in the language of algebraic stacks by noting thatfor linearly reductive groups, a GIT quotient
X//G can be interpreted as the goodmoduli space of the quotient stack [ X ss /G ]. The purpose of this paper is to give acomplete generalization of Kirwan’s result to algebraic stacks.Precisely, we prove (Theorem 2.11) that if X is a (not necessarily smooth) Artin stackwith stable good moduli space X π → X , then there is a canonical sequence of birationalmorphisms of stacks X n → X n − . . . → X = X with the following properties: (1) If X k is connected , then the maximum dimension of a stabilizer of a point of X k +1 isstrictly smaller than the maximum dimension of a stabilizer of X k and the final stack X n has constant stabilizer dimension. (2) The morphisms X k +1 → X k induce properand birational morphisms of good moduli spaces X k +1 → X k .When the stack X is smooth, then each intermediate stack X k is smooth. Thisfollows because X k +1 is an open substack of the blowup of X k along the closed smoothsubstack X max parametrizing points with maximal dimensional stabilizer. Since X n hasconstant dimensional stabilizer it follows (Proposition A.2) that its moduli space X n has only tame quotient singularities. Thus our theorem gives a canonical procedure topartially desingularize the good moduli space X .Even in the special case of GIT quotients, our method allows us to avoid the intri-cate arguments used by Kirwan. In addition, we are not restricted to characteristic 0.However, Artin stacks with good moduli spaces necessarily have linearly reductive sta-bilizers at closed points. In positive characteristic this imposes a strong condition onthe stack. Indeed by Nagata’s theorem if G is a linearly reductive group over a field ofcharacteristic p , then G is diagonalizable and p ∤ [ G : G ].Theorem 2.11 can be combined with the destackification results of Bergh [Ber17] togive a functorial resolution of the singularities of good moduli spaces of smooth Artinstacks in arbitrary characteristic (Corollary 7.2).In the smooth case, our results were applied in [ES17] to study intersection theoryon singular good moduli spaces. There, Theorem 2.11 is used to show that the pullback A sufficient condition for all of the X k to be connected is that X is irreducible. ANONICAL REDUCTION OF STABILIZERS OF ARTIN STACKS 3 A ∗ op ( X ) Q → A ∗ ( X ) Q is injective, where A ∗ op denotes the operational Chow ring definedby Fulton [Ful84].When X is singular, but possesses a virtual smooth structure, a variant of Theorem2.11 can be applied to define numerical invariants of the stack. In the papers [KL13,KLS17] the authors use a construction similar to Theorem 2.11 for GIT quotients todefine generalized Donaldson–Thomas invariants of Calabi–Yau three-folds. For GITquotient stacks, the intrinsic blowup of [KL13, KLS17] is closely related to the saturatedblowup (Definition 3.2) of a stack X along the locus of maximal dimensional stabilizer(see 8.7). Similar ideas are also being considered in recent work in progress of Joyceand Tanaka. Outline of the proof of Theorem 2.11.
The key technical construction in our proofis the saturated blowup of a stack along a closed substack.
Saturated Proj and blowups. If X is an Artin stack with good moduli space morphism X π → X , then the blowup of X along C does not necessarily have a good moduli space.The reason is that if A is any sheaf of graded O X -modules, then Proj X ( A ) need not havea good moduli space. However, we prove (Proposition 3.4) that there is a canonical opensubstack Proj π X ( A ) ⊂ Proj X ( A ) whose good moduli space is Proj X ( π ∗ A ). In general themorphism Proj π X ( A ) → X is not proper, but the natural morphism Proj X ( π ∗ A ) → X is identified with the morphism of good moduli spaces induced from the morphism ofstacks Proj π X ( A ) → X . We call Proj π X ( A ) the saturated Proj of A relative to the goodmoduli space morphism π (Definition 3.1).If C is a closed substack of X with sheaf of ideals I , then we call Bl π C X := Proj π X (cid:0)L I n (cid:1) the saturated blowup of X along C . When X and C are smooth, then Bl π C X has a partic-ularly simple description (Proposition 4.5). It is the complement of the strict transformof the saturation of C with respect to the good moduli space morphism π : X → X .Given a closed substack C ⊂ X the
Reichstein transform R ( X , C ) of X along C isthe complement of the strict transform of the saturation of C in the blowup Bl C X .The Reichstein transform was introduced in [EM12] where toric methods were used toprove that there is a canonical sequence of toric Reichstein transforms, called stackystar subdivisions, which turn an Artin toric stack into a Deligne–Mumford toric stack.The term “Reichstein transform” was inspired by Reichstein’s paper [Rei89] whichcontains the result that if C ⊂ X is a smooth, closed G -invariant subvariety of a smooth, G -projective variety X then (Bl C X ) ss is the complement of the strict transform of thesaturation of C ∩ X ss in the blowup of X ss along C ∩ X ss . Outline of the proof when X is smooth and connected. If X is a smooth Artin stackwith good moduli space X → X , then the substack X max , corresponding to pointswith maximal dimensional stabilizer, is closed and smooth. Thus X ′ = R ( X , X max )is a smooth Artin stack whose good moduli space X ′ maps properly to X and is anisomorphism over the complement of X max , the image of X max in X . The stabilityhypothesis ensures that as long as the stabilizers are not all of constant dimension, X max is a proper closed substack of X . Using the local structure theorem of [AHR15] D. EDIDIN AND D. RYDH we can show (Proposition 6.1) that the maximum dimension of the stabilizer of a pointof X ′ is strictly smaller than the maximum dimension of the stabilizer of a point of X .The proof then follows by induction.In the local case, Proposition 6.1 follows from Theorem 5.1 which states that if G is a connected, linearly reductive group acting on a smooth affine scheme, thenthe equivariant Reichstein transform R G ( X, X G ) has no G -fixed points. The proof ofTheorem 5.1 is in turn reduced to the case that X = V is a representation of G , wherethe statement can be checked by direct calculation (Proposition 5.4). The general case.
For a singular stack X , the strategy is essentially the same as inthe smooth case. The locus of points X max of maximum dimensional stabilizer has acanonical substack structure but this need not be reduced. The proof is a bit techni-cal, particularly in positive characteristic, and is given in the Appendix. When X issingular, the Reichstein transform R ( X , X max ) is not so useful: the maximum stabi-lizer dimension need not drop (cf. Example 5.7) and the Reichstein transform need notadmit a good moduli space (cf. Example 5.8).The saturated blowup, however, always admit a good moduli space. Moreover, weprove that if X max = X and X is connected, then the saturated blowup Bl π X max X has strictly smaller dimensional stabilizers. This is again proved by reducing to thecase that X = [ X/G ] where X is an affine scheme. Since, X can be embedded into arepresentation V of G , we can use the corresponding result for smooth schemes andfunctorial properties of saturated blowups (Proposition 3.11) to prove the result. Conventions and Notation.
All algebraic stacks are assumed to have affine diagonaland be of finite type over an algebraically closed field k .A point of an algebraic stack X is an equivalence class of morphisms Spec K x → X where K is a field, and ( x ′ , K ′ ) ∼ ( x ′′ , K ′′ ) if there is a k -field K containing K ′ , K ′′ such that the morphisms Spec K → Spec K ′ x ′ → X and Spec K → Spec K ′′ x ′′ → X areisomorphic. The set of points of X is denoted |X | .Since X is of finite type over a field it is noetherian. This implies that every point of ξ ∈ |X | is algebraic [LMB00, Th´eor`eme 11.3], [Ryd11, Appendix B]. This means that ifSpec K x → X is a representative for ξ , then the morphism x factors as Spec K x → G ξ →X , where x is faithfully flat and G ξ → X is a representable monomorphism. Moreover, G ξ is a gerbe over a field k ( ξ ) which is called the residue field of the point ξ . Thestack G ξ is called the residual gerbe and is independent of the choice of representativeSpec K x → X .Given a morphism Spec K x → X , define the stabilizer group G x to be the fiberproduct: G x (cid:15) (cid:15) / / Spec K ( x,x ) (cid:15) (cid:15) X ∆ X / / X × k X ANONICAL REDUCTION OF STABILIZERS OF ARTIN STACKS 5
Since the diagonal is representable G x is a K -group which we call the stabilizer of x .Since we work over an algebraically closed field, any closed point is geometric and isrepresented by a morphism Spec k x → X . In this case the residual gerbe is BG x where G x is the stabilizer of x .2. Stable good moduli spaces and statement of the main theorem
Stable good moduli spaces.Definition 2.1 ([Alp13, Definition 4.1]) . A morphism π : X → X from an algebraicstack to an algebraic space is a good moduli space if(1) π is cohomologically affine , meaning that the pushforward functor π ∗ on thecategory of quasi-coherent O X -modules is exact.(2) The natural map O X → π ∗ O X is an isomorphism.More generally, a morphism of Artin stacks φ : X → Y satisfying conditions (1) and (2)is called a good moduli space morphism . Remark 2.2.
The morphism π is universal for maps to algebraic spaces, so the alge-braic space X is unique up to isomorphism [Alp13, Theorem 6.6]. Thus, we can referto X as the good moduli space of X . Remark 2.3. If X → X is a good moduli space, then the stabilizer of any closed pointof X is linearly reductive by [Alp13, Proposition 12.14]. Remark 2.4.
Let X be a stack with finite inertia I X → X . By the Keel–Mori theorem,there is a coarse moduli space π : X → X . Following [AOV08] we say that X is tame if π is cohomologically affine. This happens precisely when the stabilizer groups are linearlyreductive. In this case X is also the good moduli space of X by [Alp13, Example 8.1].Conversely, if π : X → X is the good moduli space of a stack such that all the stabilizersare 0-dimensional, then X is a tame stack with coarse moduli space X (PropositionA.1). More generally, if the stabilizers of X have constant dimension n and X is reduced,then X is a gerbe over a tame stack whose coarse space is X (Proposition A.2). Definition 2.5.
Let π : X → Y be a good moduli space morphism. A point x of X is stable relative to π if π − ( π ( x )) = { x } under the induced map of topologicalspaces |X | → |Y | . A point x of X is properly stable relative to π if it is stable anddim G x = dim G π ( x ) .We say π is a stable (resp. properly stable) good moduli space morphism if the setof stable (resp. properly stable) points is dense.The dimension of the fibers of the relative inertia morphism Iπ → X is an uppersemi-continuous function [SGA3, Expos´e VIb, Proposition 4.1]. Hence the set X >d = { x ∈ |X | : dim G x − dim G π ( x ) > d } is closed. Proposition 2.6.
The set of stable points defines an open (but possibly empty) substack X s ⊂ X which is saturated with respect to the morphism π . If X is irreducible with D. EDIDIN AND D. RYDH generic point ξ , then X s = X r π − ( π ( X >d )) where d = dim G ξ − dim G π ( ξ ) is the minimum dimension of the relative stabilizergroups. In particular, dim G x − dim G π ( x ) = d at all points of X s .Proof. If Y → Y is any morphism, then ( Y × Y X ) s = Y × Y X s . Indeed, if x is stable,then G x → G π ( x ) is a good moduli space [Alp13, Lemma 4.14], hence a gerbe, so π − ( π ( x )) → π ( x ) is universally injective. We can thus reduce to the case where Y = X is a scheme.If Z is an irreducible component of X , then the map Z π | Z → π ( Z ) is a good modulispace morphism by [Alp13, Lemma 4.14].If x is a point of X , then π − ( π ( x )) = S Z⊂X ( π | Z ) − ( π ( x )) where the union is overthe irreducible components of X which contain x . Thus a point x is stable if and only if( π | Z ) − ( π | Z ( x )) = x for every irreducible component Z containing x . If we let Z s be theset of stable points for the good moduli space morphism π | Z , then X s = ( S Z ( Z r Z s )) c where the union is over all irreducible components of X . Since we assume that X isnoetherian there are only a finite number of irreducible components. Thus, it sufficesto prove that Z s is open for each irreducible component Z . In other words, we arereduced to the case that X is irreducible.Then X ≤ d = X d = { x ∈ |X | : dim G x = d } is open and dense and to see that X s = ( π − ( π ( X >d ))) c we argue as follows.By [Alp13, Proposition 9.1] if x is a point of X and π − ( π ( x )) is not a singleton, then π − ( π ( x )) contains a unique closed point y and dim G y is greater than the dimensionof any other stabilizer in π − ( π ( x )). Such a point is clearly not in the open set X d , sowe conclude that ( X s ) c ⊂ π − ( π ( X >d )) or equivalently that X s ⊃ ( π − ( π ( X >d ))) c .To obtain the reverse inclusion we need to show that if x is a point of X and π − ( π ( x )) = x , then dim G x = d . Consider the stack π − ( π ( x )) with its reduced stackstructure. The monomorphism from the residual gerbe G x → X factors through amonomorphism G x → π − ( π ( x ))). Since π − ( π ( x )) has a single point the morphism G x → π − ( π ( x )) red is an equivalence [Sta16, Tag 06MT]. Hence dim π − ( π ( x )) =dim G x = − dim G x .Let ξ be the unique closed point in the generic fiber of π . Then x ∈ { ξ } so byupper semi-continuity dim G x ≥ dim G ξ and dim π − ( π ( x )) ≥ dim π − ( π ( ξ )). More-over, dim π − ( π ( ξ )) ≥ − dim G ξ with equality if and only if π − ( π ( ξ )) is a singleton. Itfollows that dim π − ( π ( ξ )) ≥ − dim G ξ ≥ − dim G x = dim π − ( π ( x ))is an equality so the generic fiber π − ( π ( ξ )) is a singleton and dim G x = dim G ξ = d . (cid:3) Let X be a reduced and irreducible Artin stack and let π : X → X be a good modulispace morphism with X an algebraic space and let X s = π ( X s ). Since X s is saturated, X s is open in X . ANONICAL REDUCTION OF STABILIZERS OF ARTIN STACKS 7
Proposition 2.7.
With notation as in the preceding paragraph X s is a gerbe over atame stack with coarse space X s . Moreover, X s is the largest saturated open substackwith this property.Proof. By Proposition 2.6 the dimension of the stabilizer G x is constant at every point x of X s . Hence by Proposition A.2, X s is a gerbe over a tame stack whose coarse spaceis X s .Conversely, if U is a saturated open substack which is a gerbe over a tame stack U tame ,then the good moduli space morphism U → U factors via U tame . Since |U | → |U tame | and |U tame → U | are homeomorphisms, it follows that U ⊂ X s by definition. (cid:3) Examples.Remark 2.8. If X = [ X/G ] is a quotient stack with X = Spec A an affine variety and G is a linearly reductive group, then the good moduli space morphism X → X = Spec A G is stable if and only if the action is stable in the sense of [Vin00]. This means that thereis a closed orbit of maximal dimension. The morphism X → X is properly stable if themaximal dimension equals dim G . Following [Vin00] we will say that a representation V of a linearly reductive group G is stable if the action of G on V is stable. Example 2.9. If X = A and G = G m acts on X by λ ( a, b ) = ( λa, b ), then the goodmoduli space morphism [ X/G ] → A is not stable since the inverse image of everypoint under the quotient map A → A , ( a, b ) b contains a point with stabilizer ofdimension 1. On the other hand, if we consider the action of G m given by λ ( a, b ) =( λ d a, λ − e b ) with d, e >
0, then the good moduli space morphism [
X/G ] → A isproperly stable, since the inverse image of the open set A r { } is the Deligne–Mumfordsubstack [( A r V ( xy )) / G m ]. Example 2.10.
Consider the action of GL n on gl n via conjugation in characteristiczero. If we identify gl n with the space A n of n × n matrices, then the map gl n → A n which sends a matrix to the coefficients of its characteristic polynomial is a goodquotient, so the map π : [ gl n / GL n ] → A n is a good moduli space morphism. The orbitof an n × n matrix is closed if and only if it is diagonalizable. Since the stabilizer ofa matrix with distinct eigenvalues is a maximal torus T , such matrices have orbits ofdimension n − n = dim GL n − dim T which is maximal.If U ⊂ A n is the open set corresponding to polynomials with distinct roots, then π − ( U ) is a T -gerbe over the scheme U . Hence π is a stable good moduli space mor-phism, although it is not properly stable.2.3. Statement of the main theorem.Theorem 2.11.
Let X be an Artin stack with stable good moduli space X π → X and let E ⊂ X be an effective Cartier divisor (possibly empty). There is a canonical sequenceof birational morphisms of Artin stacks X n → X n − . . . → X = X , closed substacks ( C ℓ ⊂ X ℓ ) ≤ ℓ ≤ n − , and effective Cartier divisors ( E ℓ ⊂ X ℓ ) ≤ ℓ ≤ n , E = E , with thefollowing properties for each ℓ = 0 , , . . . , n − . D. EDIDIN AND D. RYDH (1a) |C ℓ | is the locus in X ℓ r X sℓ of points of maximal dimensional stabilizer. (1b) E ℓ +1 is the inverse image of C ℓ ∪ E ℓ . (1c) If X is smooth and E is snc, then C ℓ +1 and X ℓ +1 are smooth and E ℓ +1 is snc. (2a) There is a stable good moduli space π ℓ +1 : X ℓ +1 → X ℓ +1 . If π is properly stable,then so is π ℓ +1 . (2b) The morphism f ℓ : X ℓ +1 → X ℓ induces an isomorphism X ℓ +1 r f − ℓ ( C ℓ ) → X ℓ r π − ℓ (cid:0) π ℓ ( C ℓ ) (cid:1) . In particular, we have an isomorphism X sℓ +1 r E ℓ +1 → X sℓ r E ℓ . (2c) The morphism X ℓ +1 → X ℓ induces a projective birational morphism of goodmoduli spaces X ℓ +1 → X ℓ . (3) The maximum dimension of the stabilizers of points of X ℓ +1 r X sℓ +1 is strictlysmaller than the maximum dimension of the stabilizers of points of X ℓ r X sℓ .The final stack X n has the following properties: (4a) Every point of X n is stable. In particular, π n is a homeomorphism and thedimension of the stabilizers of X n is locally constant. (4b) X n → X is an isomorphism over X s and X n r E n = X s r E . In particular, X s ⊂ X n is schematically dense. (4c) If X is properly stable, then X n is a tame stack and X n its coarse moduli space. (4d) If X s r E is a gerbe over a tame stack (e.g., if X s is reduced), then X n is agerbe over a tame stack.The tame stack above is separated if and only if X is separated. The sequence X n →X n − → · · · → X does not depend on E . Remark 2.12.
The birational morphisms X ℓ +1 → X ℓ are Reichstein transforms in thecenters C ℓ if X is smooth and saturated blowups in the centers C ℓ in general. They arediscussed in the next section. The closed substack C ℓ is the set of points in X ℓ r X sℓ withmaximal-dimensional stabilizer equipped with a canonical scheme structure which neednot be reduced. In particular, C ℓ ∩X sℓ = ∅ and X n → X is an isomorphism over the stablelocus. Note that if X ℓ is connected, then no point with maximal dimensional stabilizercan be stable, so C ℓ is supported on the locus of points with maximal dimensionalstabilizer in X ℓ . Remark 2.13.
For singular X , there are other possible sequences X n → X n − →· · · → X that satisfy the conclusions: (1a), (2a)–(2c), (3), (4a) and (4c) but for which f − ℓ ( C ℓ ) is not a Cartier divisor. In our sequence, using saturated blowups, X s ⊂ X n is schematically dense and there are no new irreducible components appearing in theprocess. It is, however, possible to replace the saturated blowups with variants such assaturated symmetric blowups or intrinsic blowups, see Remark 8.7. For these variants, X s is typically not schematically dense and X n have additional irreducible components.3. Saturated Proj and saturated blowups
In this section we study a variant of Proj and blowups that depends on a morphism π .In every result, π will be a good moduli space morphism but we define the constructionsfor more general morphisms. ANONICAL REDUCTION OF STABILIZERS OF ARTIN STACKS 9
Definition 3.1 (Saturated Proj) . Let π : X → Y be a morphism of algebraic stacksand let A be a (positively) graded sheaf of finitely generated O X -algebras. Let π − π ∗ A + denote the image of the natural homomorphism π ∗ π ∗ A + → A + → A . Define Proj π X A =Proj X A r V ( π − π ∗ A + ). We call Proj π X A the saturated Proj of A relative to the mor-phism π .Note that the morphism Proj π X A → X need not be proper. Also note that there isa canonical morphism Proj π X A →
Proj Y π ∗ A : we are exactly removing the locus whereProj X A Proj Y π ∗ A is not defined. Definition 3.2 (Saturated blowups) . Let π : X → Y be a morphism of algebraicstacks and let
C ⊂ X be a closed substack with sheaf of ideals I . We let Bl π C X =Proj π X (cid:0)L I n (cid:1) and call it the saturated blowup of X in C . The exceptional divisor of thesaturated blowup is the restriction of the exceptional divisor of the blowup along theopen substack Bl π C X ⊂ Bl C X . Example 3.3 (Saturated blowups and GIT) . Let X = [ X ss /G ] where X ss = X ss ( L )is the set of L -semistable points for the action of a reductive group G on a smoothprojective variety X . Let C ⊂ X be a smooth, closed G -invariant subscheme and let C = [( C ∩ X ss /G )]. Let p : X ′ → X be the blowup of X along C and denote theexceptional divisor by E . Likewise let X ′ = Bl π C X where π : X →
X//G is the goodmoduli space morphism. Then by [Rei89, Theorem 2.4] and Proposition 4.5 below, X ′ = [( X ′ ) ss ( K ) /G ] where K = p ∗ L d ( − E ) for some d ≫ Proposition 3.4. If π : X → Y is a good moduli space morphism and A is a finitelygenerated graded O X -algebra, then Proj π X A →
Proj Y π ∗ A is a good moduli space mor-phism and the morphism of good moduli spaces induced by the morphism Proj π X A → X is the natural morphism
Proj Y π ∗ A → Y .Proof.
To show that the natural morphismProj π X A →
Proj Y π ∗ A is a good moduli space morphism, we may, by [Alp13, Proposition 4.9(ii)], work locallyin the smooth or fppf topology on Y and assume that Y is affine. In this case Proj π ∗ A is the scheme obtained by gluing the affine schemes Spec( π ∗ A ) ( f ) as f runs throughelements f ∈ π ∗ A + . Likewise, Proj π X A is the open set in Proj X A obtained by gluingthe X -affine stacks Spec X A ( f ) as f runs through π ∗ A + . It is thus enough to prove thatSpec X A ( f ) → Spec Y ( π ∗ A ) ( f ) is a good moduli space morphism. By [Alp13, Lemma 4.14] if A is a sheaf of coherent O X -algebras, then Spec X A →
Spec Y π ∗ A is a good moduli space morphism and the diagramSpec X A / / (cid:15) (cid:15) X π (cid:15) (cid:15) Spec Y π ∗ A / / Y is commutative. Since good moduli space morphisms are invariant under base change[Alp13, Proposition 4.9(i)] we see thatSpec X A r V ( π − ( π ∗ A + )) → Spec Y π ∗ A r V ( π ∗ A + )is a good moduli space morphism. Now Proj Y π ∗ A is the quotient of Spec Y π ∗ A r V ( π ∗ A + ) by the action of G m on the fibers over Y . It is a coarse quotient since π ∗ A isnot necessarily generated in degree 1. Likewise, Proj π X A is the quotient of Spec X A r V ( π − ( π ∗ A + )) by the action of G m on the fibers over X .Since the property of being a good moduli space is preserved by base change,Spec X A f → Spec( π ∗ A ) f is a good moduli space morphism. This gives us the com-mutative diagram Spec X A f q X / / π A f (cid:15) (cid:15) Spec X A ( f ) π A ( f ) (cid:15) (cid:15) Spec Y ( π ∗ A ) f q Y / / Spec Y ( π ∗ A ) ( f ) where π A f is a good moduli space morphism and q X and q Y are coarse G m -quotients.Note that the natural transformation M → ( q ∗ q ∗ M ) is an isomorphism for q = q X and q = q Y . Since ( π A ) ∗ is compatible with the grading, it follows that( π A ( f ) ) ∗ M = (( q Y ) ∗ ( π A f ) ∗ ( q X ) ∗ M ) is a composition of right-exact functors, hence exact. It follows that π A ( f ) is a goodmoduli space morphism. (cid:3) Variation of GIT.Remark 3.5.
Let f : X ′ → X be a projective morphism and let π : X → Y be a goodmoduli space morphism. Choose an f -ample line bundle L . Then X ′ = Proj X L n ≥ f ∗ L n .We obtain an open substack X ′L := Proj π X L n ≥ f ∗ L n of X ′ and a good moduli spacemorphism X ′L → Proj Y M n ≥ π ∗ f ∗ L n . The open substack X ′L is the locus where f ∗ π ∗ π ∗ f ∗ L n → L n is surjective for all suf-ficiently divisible n , and this typically depends on L , see Example 3.6. This can beinterpreted as variation of GIT on the level of stacks. ANONICAL REDUCTION OF STABILIZERS OF ARTIN STACKS 11
Example 3.6.
Let X = B G m and f : X ′ = Proj X ( O X ⊕ V a ) → X where a > V isthe tautological line bundle on B G m . Then X ′ has three points: two closed points P and P corresponding to the projections O X ⊕ V a → O X and O X ⊕ V a → V a and one openpoint in their complement. Let O (1) be the tautological f -ample line bundle and let L = O ( i ) ⊗ f ∗ V j with i ≥ j ∈ Z . Then X ′L = Proj π X (cid:0) ⊕ k ≥ Sym ik ( O X ⊕ V a ) ⊗ V jk (cid:1) and: X ′L = X ′ r P if j/i = 0 X ′ r P if j/i = − a X ′ r { P , P } if − a < j/i < ∅ if j/i > j/i < − a. Properties of saturated Proj and saturated blowups.Proposition 3.7.
Let π : X → Y be a good moduli space morphism and let A be agraded finitely generated O X -algebra. (1) If A → B is a surjection onto another graded O X -algebra, then Proj π X B =Proj π X A ×
Proj X A Proj X B . In particular, there is a closed immersion Proj π X B →
Proj π X A . (2) If g : Y ′ → Y is any morphism and f : X ′ = X × Y Y ′ → X is the pull-backwith good moduli space π ′ : X ′ → Y ′ , then Proj π ′ X ′ f ∗ A = Proj π X A × X X ′ as opensubstacks of Proj X ′ f ∗ A .Proof. (1) We have a closed immersion Proj X B →
Proj X A and the saturated Proj’sare the complements of V ( π − π ∗ B + ) and V ( π − π ∗ A + ) respectively. Since π is coho-mologically affine, π ∗ preserves surjections. It follows that π − π ∗ A + → π − π ∗ B + issurjective and the result follows.(2) There is an isomorphism Proj X ′ f ∗ A = Proj X A × X X ′ and the saturated Proj’sare the complements of V ( π ′− π ′∗ f ∗ A + ) and V ( f − π − π ∗ A + ) respectively. These areequal since f ∗ π ∗ π ∗ = π ′∗ g ∗ π ∗ = π ′∗ π ′∗ f ∗ [Alp13, Proposition 4.7]. (cid:3) Proposition 3.8.
Let π : X → Y be a good moduli space morphism and let
C ⊂ X be aclosed substack with sheaf of ideals I . Let f : X ′ = Bl π C X → X be the saturated blowupwith good moduli space morphism π ′ : X ′ → Y ′ and exceptional divisor E ⊂ X ′ . (1) There exists a positive integer d such that L n ≥ π ∗ ( I dn ) is generated in degree . (2) For d as above, Y ′ = Bl π ∗ ( I d ) Y and π ′− ( F ) = d E where F is the exceptionaldivisor of Y ′ → Y . (3) As closed subsets, E = f − ( C ) and f − (cid:0) π − ( π ( C )) (cid:1) , coincide. (4) The map f induces an isomorphism X ′ r E → X r π − (cid:0) π ( C ) (cid:1) . In particular, X r π − (cid:0) π ( C ) (cid:1) ⊂ X ′ is schematically dense.Proof. (1) By Proposition 3.4, Y ′ = Proj Y π ∗ A where A = L I n . Since π ∗ A is afinitely generated algebra [AHR15, Lemma A.2], it is generated in degrees ≤ m forsome m . If d is a multiple of the degrees of a set of generators, e.g., d = m !, then π ∗ A ( d ) = L n ≥ π ∗ I dn is generated in degree 1. (2) It follows from (1) that Y ′ = Proj Y π ∗ A ( d ) = Bl π ∗ ( I d ) Y . To verify that π ′− ( F ) = d E , we may work locally on Y and at the chart corresponding to an element f ∈ Γ( X , I d ). Then X ′ = Spec X ( L n ≥ I nd ) ( f ) , Y ′ = Spec Y ( L n ≥ π ∗ I nd ) ( f ) , F = V ( f ) and d E = V ( f ).(3) follows immediately from (2) since f − (cid:0) π − ( π ( C )) (cid:1) = π ′− ( F ) as sets.(4) Since the saturated blowup commutes with flat base change on Y , the map f : X ′ → X becomes an isomorphism after restricting to X r π − (cid:0) π ( C ) (cid:1) . But f − ( π − (cid:0) π ( C ) (cid:1) = E by (3). (cid:3) Remark 3.9.
Proposition 3.8 (2) generalizes [Kir85, Lemma 3.11] via Example 3.3.Let π : X → Y be a good moduli space morphism, let
C ⊂ X be a closed substack andconsider the saturated blowup p : Bl π C X → X . We have seen that it is an isomorphismoutside X r π − (cid:0) π ( C ) (cid:1) and that X r π − (cid:0) π ( C ) (cid:1) is schematically dense in the saturatedblowup. Definition 3.10 (Strict transform of saturated blowups) . If Z ⊂ X is a closed sub-stack, then we let the strict transform of Z along p denote the schematic closure of Z r π − (cid:0) π ( C ) (cid:1) in p − ( Z ). Similarly, if f : X ′ → X is any morphism, then the stricttransform of f along p is the schematic closure of X ′ r f − π − (cid:0) π ( C ) (cid:1) in X ′ × X Bl π C X . Proposition 3.11.
Let π : X → Y be a good moduli space morphism, let
C ⊂ X be aclosed substack and let p : Bl π C X → X be the saturated blowup. (1) If Z ⊂ X is a closed substack, then Bl π C∩Z Z is the strict transform of Z along p . In particular, there is a closed immersion Bl π C∩Z
Z → Bl π C X . (2) If g : Y ′ → Y is any morphism, and X ′ = X × Y Y ′ and C ′ = C × X X ′ , then Bl π ′ C ′ X ′ is the strict transform of X ′ → X along p . In particular, there is aclosed immersion Bl π ′ C ′ X ′ → Bl π C X × X X ′ and this is an isomorphism if g isflat.Proof. (1) Let A = L I n and A ′ = L I n / ( I n ∩ J ) where I defines C and J defines Z . Then there is a closed immersion Bl π C∩Z Z = Proj π X A ′ → Proj π X A = Bl π C X byProposition 3.7 (1). The result follows since Bl π C∩Z Z equals p − ( Z ) outside π − (cid:0) π ( C ) (cid:1) and is schematically dense.(2) Let A = L I n and A ′ = L I ′ n where I defines C and I ′ defines C ′ . Then f ∗ A →A ′ is surjective, so there is a closed immersion Bl π ′ C ′ X ′ → Bl π C X × X X ′ (Proposition 3.7).The result follows, since Bl π ′ C ′ X ′ → X ′ is an isomorphism outside π − (cid:0) π ( C ′ ) (cid:1) and X ′ r π − (cid:0) π ( C ′ ) (cid:1) is schematically dense in Bl π ′ C ′ X ′ (Proposition 3.8 (4)). (cid:3) Proposition 3.12.
Let π : X → Y be a good moduli morphism and let A be a finitelygenerated graded O X -algebra. Let f : X ′ := Proj π X A → X be the saturated Proj and let π ′ : X ′ → Y ′ := Proj Y π ∗ A be its good moduli space morphism. (1) If π is properly stable, then π ′ is properly stable. (2) If π is stable and A = L n ≥ I n for an ideal I , then π ′ is stable. ANONICAL REDUCTION OF STABILIZERS OF ARTIN STACKS 13
More precisely, in (1), or in (2) under the additional assumption that X is reduced,the inclusion X ′ ⊂ Proj X A is an equality over X s and X ′ s contains f − ( X s ) . In (2), X ′ s always contains f − ( X s r V ( I )) .Proof. The question is smooth-local on Y so we can assume that Y is affine. We canalso replace Y with π ( X s ) and assume that X = X s , that is, every stabilizer of X hasthe same dimension.In the first case, π is a coarse moduli space. The induced morphism π A : Spec X A →
Spec Y π ∗ A is then also a coarse moduli space. The image along π A of V ( A + ) is V ( π ∗ A + ). Since π A is a homeomorphism, √ π − π ∗ A + = √A + . It follows that X ′ =Proj X A .In the second case, if in addition X is reduced, then π factors through a gerbe g : X → X tame and a coarse moduli space h : X tame → Y (Proposition A.2). Since I n = g ∗ g ∗ I n , we conclude that Proj π X A = (cid:0) Proj h X tame g ∗ A (cid:1) × X tame X and the questionreduces to the first case.In the second case, without the additional assumption on X , let U := X s r V ( I ).Then Proj X A → X is an isomorphism over U and Y ′ := Proj Y π ∗ A → Y is anisomorphism over π ( U ) so U ⊂ X ′ s . Moreover, U ⊂ X ′ is dense so X ′ is stable. (cid:3) The condition that A is a Rees algebra in (2) is not superfluous. In Example 3.6( a = 1, i = 1, j = 0), we have a stable, but not properly stable, good moduli space π : X = B G m → Y = Spec k and a saturated Proj X ′ → X such that X ′ is not stable: X ′ = [ A / G m ] → Y ′ = Y = Spec k . Remark 3.13 (Deligne–Mumford stacks) . Proposition 3.4 is a non-trivial statementeven when X is Deligne–Mumford. In this case, the saturated Proj coincides with theusual Proj, see proof of Proposition 3.12. We can thus identify the coarse space of ablowup along a sheaf of ideals I as Proj( ⊕ π ∗ I k ) (Proposition 3.4) and as the blowupin π ∗ I d for sufficiently divisible d (Proposition 3.8). Example 3.14.
Let X = [ A /µµµ ] where µµµ acts by − ( a, b ) = ( − a, − b ). The coarsespace of X is the cone Y = Spec[ x , xy, y ]. Proposition 3.4 says that if we let X ′ bethe blowup of A at the origin, then the quotient X ′ /µµµ is Proj of the graded ring ⊕ S i where S i is the monomial ideal in the invariant ring k [ x , xy, y ] generated bymonomials of degree ⌈ i/ ⌉ . This is isomorphic to the blowup of Y in ( x , xy, y ). Remark 3.15 (Adequate stacks) . Proposition 3.4 also holds for stacks with adequatemoduli spaces with essentially identical arguments.3.3.
Resolutions of singularities of stacks with good moduli spaces.
Recallthat in characteristic zero, we have functorial resolution of singularities by blowups insmooth centers [BM08, Thm. 1.1]. To be precise, there is a functor BR which produces,for each reduced scheme X of finite type over a field of characteristic zero, a resolution ofsingularities BR ( X ) which commutes with smooth morphisms. Here BR ( X ) = { X ′ →· · · → X } is a sequence of blowups in smooth centers with X ′ smooth. Also see [Kol07, Theorem 3.36] although that algorithm may involve centers that are singular [Kol07,Example 3.106].Artin stacks can be expressed as quotients [
U/R ] of groupoid schemes s, t : R / / / / U with s and t smooth morphisms. Thus, the resolution functor BR extends uniquely toArtin stacks. In particular, for every reduced Artin stack X of finite type over a fieldof characteristic zero, there is a projective morphism ˜ X → X , a sequence of blowups,which is an isomorphism over a dense open set. Similarly, if X is a scheme with anaction of a group scheme G , then there is a sequence of blowups in G -equivariantsmooth centers that resolves the singularities of X .In general if X is an Artin stack, then a resolution of singularities ˜ X need not have agood moduli space. However, the theory of saturated blowups implies that ˜ X containsan open set which has a good moduli space such that the induced map of good modulispaces is proper and birational. Proposition 3.16.
Let X be an integral Artin stack with stable good moduli spacemorphism X π → X . Suppose that ˜ X → X is a projective birational morphism. Furtherassume that either (1) X is properly stable, or (2) ˜ X → X is a sequence of blowups.Then there exists an open substack X ′ ⊂ ˜ X such that X ′ has a stable good moduli space X ′ → X ′ and the induced morphism of good moduli spaces is projective and birational.Proof. Since ˜
X → X is projective we can write ˜ X = Proj X A for some graded sheaf A of finitely generated O X -algebras. If ˜ X → X is a blowup, we choose A as the Reesalgebra of this blowup. We treat a sequence of blowups by induction.Let X ′ = Proj π X A . By Proposition 3.4, X ′ → X ′ = Proj X π ∗ A is a good modulispace morphism. By Proposition 3.12 it is stable. If ˜ X → X is an isomorphism overthe open dense subset U ⊂ X (resp. a sequence of blowups with centers outside U ),then X ′ → X is an isomorphism over the open dense subset π ( U ∩ X s ). (cid:3) Corollary 3.17.
Let X be an integral Artin stack with stable good moduli space X π → X defined over a field of characteristic 0. There exists a quasi-projective birationalmorphism X ′ → X with the following properties. (1) The stack X ′ is smooth and admits a good moduli space X ′ π ′ → X ′ . (2) The induced map of moduli spaces X ′ → X is projective and birational.Proof. Follows immediately from functorial resolution of singularities by a sequence ofblowups, and Proposition 3.16. (cid:3) Reichstein transforms and saturated blowups
When X and C are smooth, then the saturated blowup of X along C has a particularlynice description in terms of Reichstein transforms . ANONICAL REDUCTION OF STABILIZERS OF ARTIN STACKS 15
The following definition is a straightforward extension of the one originally made in[EM12].
Definition 4.1.
Let X π → Y be a good moduli space morphism and let C ⊂ X be aclosed substack. The
Reichstein transform with center C , is the stack R ( X , C ) obtainedby deleting the strict transform of the saturation π − ( π ( C )) in the blowup of X along C . Recall that if f : Bl C X → X is the blowup, then E = f − ( C ) is the exceptionaldivisor and f − ( Z ) − E = Bl C∩Z Z is the strict transform of Z ⊂ X . Remark 4.2.
Observe that if X and C are smooth, then R ( X , C ) is smooth since it isan open set in the blowup of a smooth stack along a closed smooth substack. Remark 4.3.
Let X ′ ψ / / π ′ (cid:15) (cid:15) X π (cid:15) (cid:15) Y ′ φ / / Y be a cartesian diagram where the horizontal maps are flat and the vertical mapsare good moduli morphisms. If C ⊂ X is a closed substack, then R ( X ′ , ψ − C ) = X ′ × X R ( X , C ). This follows because blowups commute with flat base change andthe saturation of ψ − ( C ) is the inverse image of the saturation of C . Definition 4.4 (Equivariant Reichstein transform) . If an algebraic group G acts on ascheme X with a good quotient p : X → X//G and C is a G -invariant closed subscheme,then we write R G ( X, C ) for the complement of the strict transform of p − p ( C ) in theblowup of X along C . There is a natural G -action on R G ( X, C ) and R ([ X/G ] , [ C/G ]) =[ R G ( X, C ) /G ]. Proposition 4.5.
Let π : X → Y be a good moduli space morphism and let
C ⊂ X bea closed substack. If X and C are smooth, then R ( X , C ) = Bl π C X as open substacks of Bl C X . In general, Bl π C X ⊂ R ( X , C ) .Proof. Let A = ⊕ n ≥ I n . The saturation of C is the subscheme defined by the ideal J = π ∗ I · O X so the strict transform of the saturation is the blowup of the substack V ( J ) along the ideal I / J , which is Proj C ( ⊕ n ≥ ( I n / ( I n ∩ J )). Thus the ideal of thestrict transform of the saturation is the graded ideal ⊕ n> ( I n ∩ J ) ⊂ A . We needto show that this ideal defines the same closed subset of the blowup as the ideal π − π ∗ ( A + ).Since A + = ⊕ n> I n we have that π − π ∗ ( A + ) = π ∗ ( A + ) · A = ⊕ n ≥ K n where K n = P k> π ∗ ( I k ) I n − k . We need to show that p ⊕ n> I n ∩ J = p ⊕ n> K n in A . Observe that π ∗ ( I k ) ·O X ⊂ I k and π ∗ ( I k ) ·O X ⊂ π ∗ ( I ) ·O X = J so π ∗ ( I k ) I n − k ⊂I n ∩ J . Hence K n ⊂ I n ∩ J .To establish the opposite inclusion, we work smooth-locally on Y . We may thusassume that Y = Spec A and π ∗ I = ( f , f , . . . , f a ) ⊂ A . The ideal I n ∩ J can locallybe described as all functions in J = ( f , f , . . . , f a ) · O X that vanish to order at least n along C . If ord C ( f i ) = d i , that is, if f i ∈ I d i r I d i +1 , then for any n greater than allthe d i ’s, we have that I n ∩ J = P ai =1 f i · I n − d i . Since f i ∈ π ∗ ( I d i ) it follows that I n ∩ J ⊂ a X i =1 π ∗ ( I d i ) I n − d i ⊂ K n . Thus ⊕ n> ( I n ∩ J ) ⊂ √⊕ n> K n which completes the proof. (cid:3) The following example shows that if C is singular, then the Reichstein transformneed not equal the saturated blowup and could even fail to have a good moduli space.Also see Examples 5.7 and 5.8. Example 4.6.
Let U = Spec k [ x, y ] where G m acts by λ ( a, b ) = ( λa, λ − b ) and let X = [ U/ G m ]. Its good moduli space is X = Spec k [ xy ]. Let Z = V ( x y, xy ) ⊂ U and C = [ Z/ G m ]. Its saturation is sat C = V ( x y ) which has strict transform Bl C sat C = ∅ .Thus, the Reichstein transform R ( X , C ) equals Bl C X .We will show Bl C X has no good moduli space. To see this, note that ( x y, y x ) =( xy ) · ( x, y ). Since ( xy ) is invertible we conclude that Bl C X = Bl P X where P = V ( x, y ).The exceptional divisor of the latter blowup is [ P / G m ] where G m acts by λ [ a : b ] =[ λa : λ − b ]. This has no good moduli space since the closure of the open orbit containsthe two fixed points [0 : 1] and [1 : 0].The Reichstein transform R ( X , P ), on the other hand, equals Bl P X r { [0 : 1] , [1 : 0] } which is tame with coarse moduli space Bl π ( P ) X = X .5. Equivariant Reichstein transforms and fixed points
The goal of this section is to prove the following theorem.
Theorem 5.1.
Let X = Spec A be a smooth affine scheme with the action of a con-nected linearly reductive group G . Then R G ( X, X G ) G = ∅ . Remark 5.2.
By [CGP10, Proposition A.8.10] the fixed locus X G is a closed smoothsubscheme of X . Note that if G acts trivially, then X G = X and R G ( X, X G ) = ∅ . Remark 5.3.
Theorem 5.1 is false if we drop the assumption that X is smooth. SeeExample 5.7 below.5.1. The case of a representation.
In this section we prove Theorem 5.1 when X = V is a representation of G . Proposition 5.4.
Let V be a representation of a connected linearly reductive group G .Then R G ( V, V G ) G = ∅ . ANONICAL REDUCTION OF STABILIZERS OF ARTIN STACKS 17
Proof.
Decompose V = V ⊕ V m such that V is the trivial submodule and V m is a sumof non-trivial irreducible G -modules. Viewing V as a variety we write V = V × V m .The fixed locus for the action of G on V is V × { } , so the blowup of V along V G is isomorphic to V × ˜ V m where ˜ V m is the blowup of V m at the origin. Also, thesaturation of V G is V × sat G { } where sat G { } is the G -saturation of the origin inthe representation V m . Thus R G ( V, V G ) = V × R G ( V m ,
0) so to prove the propositionwe are reduced to the case that V = V m ; that is, V is a sum of non-trivial irreduciblerepresentations and { } is the only G -fixed point.To prove the proposition we must show that every G -fixed point of the exceptionaldivisor P ( V ) ⊂ ˜ V is contained in the strict transform ofsat G { } = { v ∈ V : 0 ∈ Gv } . Let x ∈ P ( V ) be a G -fixed point. The fixed point x corresponds to a G -invariantline L ⊂ V , inducing a character χ of G . Since the origin is the only fixed point,the character χ is necessarily non-trivial. Let λ be a 1-parameter subgroup such that h λ, χ i >
0. Then λ acts with positive weight α on L and thus L ⊂ sat λ { } = V + λ ∪ V − λ where V + λ = { v ∈ V : lim t → λ ( t ) v = 0 } ,V − λ = { v ∈ V : lim t →∞ λ ( t ) v = 0 } are the linear subspaces where λ acts with positive weights and negative weights re-spectively.Since sat G { } ⊃ sat λ { } , it suffices to show that x ∈ P ( V ) lies in the strict transformof sat λ { } . The blowup of sat λ { } in the origin intersects the exceptional divisor of ˜ V in the (disjoint) linear subspaces P ( V + λ ) ∪ P ( V − λ ) ⊂ P ( V ). Since L ⊂ V + λ we see thatour fixed point x is in P ( V + λ ) as desired. (cid:3) Completion of the proof of Theorem 5.1.
The following lemma is a specialcase of [Lun73, Lemma on p. 96] and Luna’s fundamental lemma [Lun73, p. 94]. Forthe convenience of the reader, we include the first part of the proof.
Lemma 5.5 (Linearization) . Let X = Spec A be a smooth affine scheme with theaction of a linearly reductive group G . If x ∈ X G is a closed fixed point, then there is a G -saturated affine neighborhood U of x and a G -equivariant strongly ´etale morphism φ : U → T x X , with φ ( x ) = 0 . That is, the diagram U φ / / π U (cid:15) (cid:15) T x X π (cid:15) (cid:15) U//G ψ / / T x X//G is cartesian and the horizontal arrows are ´etale.
Proof.
Let m be the maximal ideal corresponding to x . Since x is G -fixed the quotientmap m → m / m is a map of G -modules. By the local finiteness of group actions thereis a finitely generated G -submodule V ⊂ m such that the restriction V → m / m is surjective. Since G is linearly reductive there is a summand W ⊂ V such that W → m / m is an isomorphism of G -modules. Since W ⊂ A we obtain a G -equivariantmorphism X → T x X = Spec(Sym( m / m )) which is ´etale at x . Luna’s fundamentallemma now gives an open saturated neighborhood U of x such that U → T x X isstrongly ´etale. (cid:3) Remark 5.6.
Using Lemma 5.5 and arguing as in the proof of Proposition 5.4, werecover the result that X G is smooth (Remark 5.2). Completion of the Proof of Theorem 5.1.
Every G -fixed point of R G ( X, X G ) lies in theexceptional divisor P ( N X G X ). To show that R G ( X, X G ) G = ∅ we can work locally ina neighborhood of a point x ∈ X G . Thus we may assume (Lemma 5.5) that there is astrongly ´etale morphism X → T x X yielding a cartesian diagram X / / (cid:15) (cid:15) T x X (cid:15) (cid:15) X//G / / T x X//G.
Hence R G ( X, X G ) can be identified with the pullback of R G ( T x X, ( T x X ) G ) along themorphism X//G → T x X//G (Remark 4.3). By Proposition 5.4, R G ( T x X, ( T x X ) G ) G = ∅ so therefore R G ( X, X G ) G = ∅ as well. (cid:3) Example 5.7.
Note that the conclusion of Theorem 5.1 is false without the assump-tion that X is smooth. Let V be the 3-dimensional representation of G = G m withweights ( − , , f = x x + x is G -homogeneous of weight 5, so thesubvariety X = V ( f ) is G -invariant. Since all weights for the G -action are non-zero X G = ( A ) G = { } .Let ˜ A be the blowup of the origin. The exceptional divisor is P ( V ) and has three G -fixed points P = [0 : 0 : 1] , P = [0 : 1 : 0] , P = [1 : 0 : 0]. The exceptional divisorof ˜ X is the projectivized tangent cone P ( C { } X ). Since X = V ( f ) is a hypersurfaceand x x is the sole term of lowest degree in f , we see that P ( C { } X ) is the subscheme V ( x x ) ⊂ P ( V ). This subvariety contains the 3 fixed points, so ˜ X has 3 fixed points.The saturation of 0 in X with respect to the G -action is ( X ∩ V ( x )) ∪ ( X ∩ V ( x , x )).The intersection of the exceptional divisor with the strict transform of X ∩ V ( x ) isthe projective subscheme V ( x , x ) whose reduction is P .The intersection of the exceptional divisor with the strict transform of X ∩ V ( x , x )is the point V ( x , x ) = P . Thus the strict transform of the saturation of 0 in X doesnot contain all of the fixed points of ˜ X . Hence R G ( X, X G ) G = ∅ .The exceptional divisor of the saturated blowup of X in the origin is P ( C { } X ) r (cid:0) V ( x ) ∪ V ( x , x ) (cid:1) = V ( x ) r (cid:0) V ( x ) ∪ V ( x ) (cid:1) which has no G -fixed points. ANONICAL REDUCTION OF STABILIZERS OF ARTIN STACKS 19
Example 5.8.
Consider the following variation of Example 5.7. Let V be the 5-dimensional representation of G = G m with weights ( − , , , , f = x x + x and g = x x + x are G -homogeneous of weights 5 and 6 so the subva-riety X = V ( f, g ) is G -invariant. As before we blow up the origin and the exceptionaldivisor of ˜ A is P ( V ) which has five G -fixed points. The exceptional divisor of ˜ X is P ( C { } X ) which is given by V ( x x , x x ) ⊂ P ( V ). It contains the five fixed pointsof P ( V ). The saturation of 0 in X is the union of X ∩ V ( x ) = V ( x , x , x ) and X ∩ V ( x , x , x , x ) = V ( x , x , x , x ).In particular, the exceptional divisor of R G ( X, X G ) contains the closed subscheme V ( x , x , x ) = P ( W ) where W is the 2-dimensional representation with weights (1 , P ( W ) admits no good quotient by G since the closure of the open orbit containsboth G -fixed points. It follows that R G ( X, X G ) does not admit a good quotient.The exceptional divisor of the saturated blowup of X in the origin is P ( C { } X ) r (cid:0) V ( x ) ∪ V ( x , x , x , x ) (cid:1) = V ( x , x ) r (cid:0) V ( x ) ∪ V ( x , x ) (cid:1) which has no G -fixedpoints and the saturated blowup admits a good moduli space.6. The proof of Theorem 2.11 in the smooth case
In this section we prove the main theorem in the smooth case and prove that thealgorithm is functorial with respect to strong morphisms.6.1.
Proof of Theorem 2.11.
Let X be a smooth stack with a stable good modulispace and let E be an snc divisor (e.g., E = ∅ ). Taking connected components, we mayassume that X is irreducible. By Lemma B.1, for any stack X the locus X max of pointsof X with maximal dimensional stabilizer is a closed subset of |X | . Moreover, if X is smooth, then Proposition B.2 implies that X max with its reduced induced substackstructure is also smooth. When X = [ X/G ] with X smooth and G linearly reductivethen X max = [ X G /G ] where G is the reduced identity component of G . For anarbitrary smooth stack X with good moduli space X the stack structure on X max canbe ´etale locally described as follows.If x is a closed point with stabilizer G x , then by [AHR15, Theorem 2.9, Theorem1.1] there is a cartesian diagram of stacks and good moduli spaces[ U/G x ] / / (cid:15) (cid:15) X π (cid:15) (cid:15) U//G x / / X (cid:3) where the horizontal maps are ´etale. In this setup, the inverse image of X max in [ U/G x ]is [ U G /G x ] where G is the reduced identity component of G x . See Appendix B formore details.The proof of Theorem 2.11 proceeds by induction on the maximum stabilizer di-mension. First suppose that the maximum stabilizer dimension equals the minimumstabilizer dimension. Then X s = X and X is a gerbe over a tame stack X tame by Proposition A.2. If X is properly stable, then every stabilizer has dimension zero and X = X tame is a tame stack by Proposition A.1. We have thus shown that the sequenceof length 0, that is X n = X = X , satisfies the conclusions (4a)–(4d) of the Theorem2.11.If the maximum stabilizer dimension is greater than the minimum stabilizer di-mension, we let X = X , C = X max and f : X = R ( X , C ) → X . The followingProposition shows that the conclusions in Theorem 2.11 hold for ℓ = 0. In particular,( X , E ) satisfies the hypothesis of Theorem 2.11 and the maximal stabilizer dimen-sion of X has dropped. By induction, we thus have X n → · · · → X such that theconclusions also hold for ℓ = 1 , . . . , n and Theorem 2.11 for X follows. Proposition 6.1.
Let X be a smooth irreducible Artin stack with stable (resp. properlystable) good moduli space morphism π : X → X and let E be an snc divisor on X . Let f : X ′ = R ( X , X max ) → X be the Reichstein transform. If X max = X then: (1a) C := X max is smooth and meets E with normal crossings. (1b) X ′ and f − ( C ) are smooth and E ′ := f − ( C ) ∪ f − ( E ) is snc. (2a) The stack X ′ has a good moduli space X ′ and the good moduli space morphism π ′ : X ′ → X ′ is stable (resp. properly stable). (2b) f induces an isomorphism X ′ r f − ( C ) → X r π − (cid:0) π ( C ) (cid:1) . In particular, f induces an isomorphism X ′ s r f − ( C ) → X s . (2c) The induced morphism of good moduli spaces X ′ → X is projective and anisomorphism over X r π ( C ) . In particular, it is an isomorphism over X s . (3) Every point of X ′ has a stabilizer of dimension strictly less than the maximumdimension of the stabilizers of points of X .Proof. Assertion (1a) is Proposition B.2. The Reichstein transform X ′ is an open sub-stack of Bl X max X . Assertion (1b) thus follows from the corresponding properties ofBl X max .Since X and X max are smooth, Proposition 4.5 implies that R ( X , X max ) is the sat-urated blowup of X along X max . Assertions (2a)–(2c) then follow from the propertiesof the saturated blowup (Propositions 3.8 and 3.12) and the fact that X max ⊂ X r X s by Proposition 2.6.We now prove assertion (3). By the local structure theorem [AHR15, Theorem 2.9]we may assume X = [ U/G x ]. Let G be the reduced identity component of G x . Then[ U/G x ] max = [ U G /G x ]. To complete the proof we need to show that R G x ( U, U G ) hasno G -fixed point. By Theorem 5.1 we know that R G ( U, U G ) has no G -fixed points.We will prove (3) by showing that R G x ( U, U G ) = R G ( U, U G ) as open subschemes ofthe blowup of U along U G .Consider the maps of quotients U π → U//G q → U//G x . If U = Spec A , then thesemaps are induced by the inclusions of rings A G x = ( A G ) ( G x /G ) ֒ → A G ֒ → A. ANONICAL REDUCTION OF STABILIZERS OF ARTIN STACKS 21
Since the quotient group G x /G is a finite k -group scheme, U//G = Spec A G → U//G x = Spec( A G ) ( G x /G ) is a geometric quotient.If C ⊂ U is a G x -invariant closed subset of U , then its image in U//G is ( G x /G )-invariant, so it is saturated with respect to the quotient map U//G → U//G x . Hence,as closed subsets of U , the saturations of C with respect to either the quotient map U → U//G or to U → U//G x are the same . It follows that if C ⊂ U is G x -invariant,then R G x ( U, C ) and R G ( U, C ) define the same open subset of the blowup of U along C . Since U G is G x -invariant we conclude that R G ( U, U G ) = R G x ( U, U G ) as opensubschemes of the blowup. (cid:3) Without the assumption that π : X → X is a stable good moduli space morphism,the conclusion in Proposition 6.1 that X ′ → X is birational can fail: it may happenthat the saturation of X max equals X and thus that X ′ = ∅ . The following examplesillustrate this. Example 6.2.
Let G m acts on X = A with weight 1. The structure map A → Spec k is a good quotient, so Spec k is the good moduli space of X = [ A / G m ]. The stabilizerof any point of A − { } is trivial, so X max = [ { } / G m ] and the saturation of X max isall of X . Hence R ( X , X max ) = ∅ . Example 6.3.
Here is a non-toric example. Let V = sl be the adjoint representationof G = SL ( C ). Explicitly, V can be identified with the vector space of traceless 2 × -action given by conjugation. Let V reg ⊂ V be the open set corre-sponding to matrices with non-zero determinant and set X = V reg × A . Let X = [ X/G ]where G acts by conjugation on the first factor and translation on the second factor.The map of affines X → A r { } given by ( A, v ) det A is a good quotient, so π : X → A r { } is a good moduli space morphism. However, the morphism π is notstable because the only closed orbits are the orbits of pairs ( A, A, v ) with v = 0 is trivial and the stabilizer of ( A,
0) isconjugate to T = diag ( t t − ) and X max = [( V reg × { } ) /G ]. Thus, π − ( π ( X max )) = X and R ( X , X max ) = ∅ .6.2. Functoriality for strong morphisms.
Let X and Y be Artin stacks with goodmoduli space morphisms, π Y : Y → Y , and π X : X → X . Let f : Y → X be a morphismand let g : Y → X be the induced morphism of good moduli spaces. The saturations with respect to the quotient maps come with natural scheme structures whichare not the same. If I ⊂ A is the ideal defining C in U , then the saturation of C with respect to thequotient map U → U//G is the ideal I G A while the ideal defining the saturation of C with respectto the quotient map U → U//G x is the ideal I G x A . While I G x A ⊂ I G A , these ideals need not beequal. Definition 6.4.
We say the morphism f is strong if the diagram Y f / / π Y (cid:15) (cid:15) X π X (cid:15) (cid:15) Y g / / X is cartesian.Note that a strong morphism is representable and stabilizer-preserving. We thushave an equality Y max = f − ( X max ) of closed substacks by Proposition B.4. A sharpcriterion for when a morphism is strong can be found in [Ryd15]. Theorem 6.5.
Let f : Y → X be a strong morphism of smooth Artin stacks withstable good moduli space morphisms
Y → Y and X → X . Let Y ′ and X ′ be the stacksproduced by Theorem 2.11. Then there is a natural morphism f ′ : Y ′ → X ′ such thatthe diagram Y ′ (cid:15) (cid:15) f ′ / / X ′ (cid:15) (cid:15) Y f / / X is cartesian.Proof. The theorem follows by induction and the following proposition.
Proposition 6.6.
Let f : Y → X be a strong morphism of smooth algebraic stacks withgood moduli spaces. Then there is a natural morphism f ′ : R ( Y , Y max ) → R ( X , X max ) such that the diagram (6.6.1) R ( Y , Y max ) (cid:15) (cid:15) f ′ / / R ( X , X max ) (cid:15) (cid:15) Y f / / X is cartesian.Proof. We will prove that f − ( X max ) = Y max , that (Bl X max X ) × X Y = Bl Y max Y andthat the open subsets R ( X , X max ) × X Y and R ( Y , Y max ) coincide. This gives a naturalmorphism f ′ such that (6.6.1) is cartesian. These claims can be verified ´etale locallyon X and Y at points of π Y ( Y max ). Let y ∈ |Y max | and x = f ( y ).Since Y → X is of finite type we can, locally around π Y ( y ), factor it as Y ֒ → X × A n → X where the first map is a closed immersion and the second map is thesmooth projection. By base change, this gives a local factorization of the morphism f as Y i ֒ → X × A n p → X .Since X × A n → X is flat and ( X × A n ) max = X max × A n it follows from Remark4.3 that R ( X × A n , ( X × A n ) max ) = R ( X , X max ) × A n . We are therefore reduced to the ANONICAL REDUCTION OF STABILIZERS OF ARTIN STACKS 23 case that the map f is a closed immersion. Since X and Y are smooth, f is necessarilya regular embedding.We can apply Theorem [AHR15, Theorem 2.9 and Theorem 1.1] to reduce to thecase that X = [ X/G ] where G = G y = G x is a linearly reductive group and X is asmooth affine scheme. Let Y = X × X Y . Then Y → X is a regular closed immersionand Y = [ Y /G ]. Since G is not necessarily smooth, it is not automatic that Y issmooth. But the fiber Y × Y BG y = Spec k is regular and Y × Y BG y → Y is a regularclosed immersion since Y is smooth. It follows that Y is smooth over an open G -invariant neighborhood of the preimage of y . After replacing Y with an open saturatedneighborhood of y [AHR15, Lemma 4.1], we can thus assume that Y is smooth.Since T x X = T y Y × N y ( Y /X ), we can slightly modify Lemma 5.5 to obtain thefollowing commutative diagram of strong ´etale morphisms (after further shrinking of X ): Y (cid:15) (cid:15) f / / X (cid:15) (cid:15) [ T y /G ] / / [ T x /G ] . Since f is strong, the action of the stabilizer G is trivial on the normal space N y . Thus( T x ) max = ( T y ) max × N y and T x //G = T y //G × N y . The result is now immediate. (cid:3) Corollaries of Theorem 2.11 in the smooth case
Reduction to quotient stacks.
Suppose that X is a smooth Artin stack suchthat the good moduli space morphism π : X → X is properly stable. The end result ofour canonical reduction of stabilizers (Theorem 2.11) is a smooth tame stack X n . Proposition 7.1.
Let X be a smooth Artin stack with properly stable good modulispace. Suppose that X n is Deligne–Mumford (automatic if char k = 0 ) and that either X has generically trivial stabilizer or X is quasi-projective. Then (1) X n is a quotient stack [ U/ GL m ] where U is an algebraic space. (2) If, in addition, X is separated, then U is separated and the action of GL m on U is proper. (3) If, in addition, X is a scheme, then so is U . (4) If, in addition, X is a separated scheme, then we can take U to be quasi-affine. (5) If, in addition, X is projective, then there is a projective variety X with alinearized action of a GL n such that X s = X ss = U . Moreover, if char k = 0 ,we can take X to be smooth.Proof. If the generic stabilizer of X is trivial, so is the generic stabilizer of X n . Henceby [EHKV01, Theorem 2.18] (trivial generic stabilizer) or [KV04, Theorem 2] (quasi-projective coarse space), X n is a quotient stack. This proves (1). If X is separated, then X n is a separated quotient stack so GL m must act properly.This proves (2). (Note that if GL m acts properly on U , then U is necessarily separated.This also follows immediately since U → X n is affine.)The morphism U → X n is affine. Indeed, there is a finite surjective morphism V → X n [EHKV01, Theorem 2.7] where V is a scheme and V → X n is finite andsurjective, hence affine. It follows that U × X n V → X n is affine and hence U → X n isaffine as well (Chevalley’s theorem). One can also deduce this directly from U → X n being representable and cohomologically affine (Serre’s theorem).In particular, if X is a scheme, then so is X n and U . This proves (3). Similarly, if X is a separated scheme, then so is X n and U . But U is a smooth separated schemeand thus has a G -equivariant ample family of line bundles. It follows that X n has theresolution property and that we can choose U quasi-affine, see [Tot04, Theorems 1.1,1.2] for further details. This proves (4).We now prove (5). Since U is quasi-affine, it is also quasi-projective. By [Sum74,Theorem 1] there is an immersion U ⊂ P N and a representation GL m → PGL N +1 suchthat the GL m -action on U is the restriction of the PGL N +1 -action on P N . Let X be theclosure of U in P N . The action of G on X is linearized with respect to the line bundle O X (1). Our statement follows from [MFK94, Converse 1.13].Finally, if char k = 0, then by equivariant resolution of singularities we can embed U into a non-singular projective G -variety X . (cid:3) Note that we only used that X n is Deligne–Mumford to deduce that X n is a quotientstack.7.2. Resolution of good quotient singularities.
Combining the main theorem withdestackification of tame stacks [Ber17, BR14], we obtain the following results, valid inany characteristic.
Corollary 7.2 (Functorial destackfication of stacks with good moduli spaces) . Let X be a smooth Artin stack with stable good moduli space morphism π : X → X . Thenthere exists a sequence X n → · · · → X → X = X of birational morphisms of smoothArtin stacks such that (1) Each X k admits a stable good moduli space π k : X k → X k . (2) The morphism X k +1 → X k is either a Reichstein transform in a smooth center,or a root stack in a smooth divisor. (3) The morphism X k +1 → X k induces a projective birational morphism of goodmoduli spaces X k +1 → X k . (4) X n is a smooth algebraic space. (5) X n → X n is a composition of a gerbe X n → ( X n ) rig and a root stack ( X n ) rig → X n in an snc divisor D ⊂ X n .Moreover, the sequence is functorial with respect to strong smooth morphisms X ′ → X ,that is, if X ′ → X is smooth and X ′ = X × X X ′ , then the sequence X ′ n → · · · → X ′ isobtained as the pull-back of X n → · · · → X along X ′ → X . ANONICAL REDUCTION OF STABILIZERS OF ARTIN STACKS 25
Proof.
We first apply Theorem 2.11 to X and can thus assume that X is a gerbe over atame stack X tame . We then apply destackification to Y := X tame . This gives a sequenceof smooth stacky blowups Y n → Y n − → · · · → Y → Y = Y , such that Y n is smoothand Y n → Y n factors as a gerbe Y n → ( Y n ) rig followed by a root stack ( Y n ) rig → Y n inan snc divisor. A smooth stacky blowup is either a root stack along a smooth divisoror a blowup in a smooth center. A blowup on a tame stack is the same thing as aReichstein transform.We let X k = X × Y Y k . Then X n → Y n → Y rig is a gerbe and X n = Y n . (cid:3) Corollary 7.3 (Resolution of good quotient singularities) . If X is a stable good modulispace of a smooth stack, then there exists a projective birational morphism p : X ′ → X where X ′ is a smooth algebraic space. The resolution is functorial with respect to smoothmorphisms. The proof of Theorem 2.11 in the singular case
Recall that the set of points |X max | ⊂ |X | which has maximal stabilizer dimensionis closed. This set has a canonical structure as a closed substack that we denote X max (Appendix B). If X is smooth, then X max is smooth. If f : X → Y is a stabilizer-preserving morphism, for example a closed immersion or a strong morphism, then X max = f − ( Y max ). When X = [ U/G ] and G has the same dimension as the maximalstabilizer dimension, then X max = [ U G /G ].The locus of stable points X s may have connected components of different stabilizerdimensions. A complication in the singular case is that the closures of these componentsmay intersect. The following lemma takes care of this problem. Lemma 8.1.
Let X be an Artin stack with stable good moduli space morphism π : X → X . Let N (resp. n ) be the maximum dimension of a stabilizer of a point of X (resp. apoint of X r X s ). Let X sk ⊂ X s denote the subset of points that are stable with stabilizerof dimension k . Let X ≤ n ⊂ X denote the subset of points with stabilizer of dimensionat most n . Then (1) X s is the disjoint union of the X sk . (2) We have a partition of X into open and closed substacks: (8.1.1) X = X ≤ n ∐ X sn +1 ∐ · · · ∐ X sN . In X ≤ n we have the following two open substacks X sn and X ∗ = X ≤ n r X sn . We let X sn and X ∗ denote their schematic closures. (3) Every point of X sn has stabilizer of dimension n and every point of X ∗ hasstabilizer of dimension at most n . (4) X sn = X ≤ n r X ∗ . Thus, every point of X r X s with stabilizer of dimension n iscontained in X ∗ . In particular, |X ∗ max | = |X r X s | max . Proof. (1) Since the stabilizer dimension is locally constant on X s , we have that X s = ` k X sk .(2) The subset of points of stabilizer dimension ≥ n + 1 is closed by upper semi-continuity. By assumption this set is also contained in X s and hence open. This givesthe decomposition in (8.1.1).(3) Every point of either X sn or X ∗ lies in X ≤ n and thus has stabilizer of dimensionat most n . Every point of X sn has stabilizer of dimension at least n by upper semi-continuity.(4) Since X s is open, X sn ⊂ X ≤ n r X ∗ . Suppose that x ∈ |X ≤ n | is not stable. If x hasstabilizer of dimension < n , then x ∈ X ∗ . If x has stabilizer of dimension n , then x isthe unique closed point in π − ( π ( x )) and there exists a generization y with stabilizerof dimension < n [Alp13, Proposition 9.1]. Thus y ∈ X ∗ and so x ∈ X ∗ . (cid:3) Remark 8.2.
If all points of X s have stabilizers of the same dimension and X 6 = X s ,then X = X ∗ = X ∗ . Two notable examples are irreducible stacks and properly stablestacks. When this is the case, the proof in the singular case simplifies quite a bit. Ingeneral, note that the closed immersion X sn ∪X ∗ → X ≤ n is surjective but not necessarilyan isomorphism if X ≤ n has embedded components. Example 8.3.
Let U = Spec k [ x, y, z ] / ( xz, yz ) and let G m act with weights (1 , − , X = [ U/ G m ] has two irreducible components: the first component X s = V ( x, y ) = X max has stabilizer G m at every point, the second component X ∗ = V ( z ) = X s has asingle point with stabilizer G m . The algorithm will blow up X ∗ max = V ( x, y, z ).We prove Theorem 2.11 in the singular case using induction on the maximum di-mension of the stabilizer of a point of X r X s and the smooth case to verify that themaximum dimension drops.First suppose X = X s and let us see that Theorem 2.11 holds with X n = X = X .Note that X has locally constant stabilizer dimensions so it is a disjoint union ofstacks with constant stabilizer dimensions. If in addition X is properly stable, then thestabilizer dimensions are all equal to zero and X is a tame stack (Proposition A.1). Ifinstead X r E is a gerbe over a tame stack (automatic if X r E is reduced by PropositionA.2), then ( X r E ) max = X r E on every connected component of X (Corollary B.7).Since X r E is schematically dense in X it follows that X max = X on every connectedcomponent of X . Thus X is a gerbe over a tame stack (Corollary B.7). We have nowproven Theorem 2.11 when X = X s .Now suppose X 6 = X s . Let n be the maximal dimension of a stabilizer of X r X s andassume that the theorem has been proven for smaller n . Let X = X and C = X ∗ max in the notation of Lemma 8.1. Let f : X = Bl π C X be the saturated blowup and E itsexceptional divisor. That the conclusion of the main theorem holds for ℓ = 0 followsby the following proposition. In particular, the maximal dimension of a stabilizer of X r X s is strictly less than n so by induction we have X n → · · · → X such that theconclusions also hold for ℓ = 1 , . . . , n and the theorem follows for X . ANONICAL REDUCTION OF STABILIZERS OF ARTIN STACKS 27
Proposition 8.4.
Let X be an Artin stack with stable good moduli space morphism π : X → X . Let n be the maximum dimension of a stabilizer of a point of X r X s .Let C = ( X ∗ ) max = ( X ∗ ) n have the scheme structure of Proposition B.4 and let X ′ =Bl π C ( X ) be the saturated blowup. Then X ′ is an Artin stack with the following properties. (2a) The stack X ′ has a good moduli space X ′ and the good moduli space morphism π ′ : X ′ → X ′ is stable (properly stable if π is properly stable). (2b) The morphism f : X ′ → X induces an isomorphism X ′ r f − ( C ) → X r π − (cid:0) π ( C ) (cid:1) . (2c) The induced morphism of good moduli spaces X ′ → X is projective and anisomorphism over the image of X s in X . (3) Every point of X ′ r ( X ′ ) s has stabilizer of dimension strictly less than n .Proof. Assertions (2a)–(2c) follow from the properties of the saturated blowup (Propo-sitions 3.8 and 3.12).Since X sn ∐ X ∗ is open in X ≤ n and its complement X ≤ n r ( X sn ∐ X ∗ ) is contained in C , we conclude that f − ( X sn ∐ X ∗ ) is schematically dense in Bl C ( X ≤ n ) and thusBl C ( X ≤ n ) = Bl C ( X sn ) ∪ Bl C ( X ∗ )and similarly for saturated blowups. We note that every point of Bl C ( X sn ) has stabilizerof dimension n by upper semi-continuity. If every point of Bl C ( X ∗ ) has stabilizer ofdimension < n , then the two components are necessarily disjoint and every point ofBl C ( X sn ) is stable. To prove (3) we may thus replace X with X ∗ . Then C = X max = X n and every stable point has dimension strictly less than n .By the local structure theorem [AHR15, Theorem 1.2] we may assume that X =[ U/G ] where G is the stabilizer of a point in X n and U = Spec A is affine. By the localfiniteness of group actions, there is a finite-dimensional G -submodule V ⊂ A such thatSym( V ) → A is surjective. We thus have a closed G -equivariant embedding U → V ∨ .Consequently, we have a closed embedding of stacks X = [ U/G ] → Y = [ V ∨ /G ]. Since X max = Y max ∩ X we obtain a closed embeddingBl X max X → Bl Y max Y . Indeed, Bl X max X is the strict transform, that is, the closure of X r X max in Bl Y max Y .This also holds for saturated blowups by Proposition 3.11.From the smooth case (Proposition 6.1 (3)), we know that Bl π Y max Y has no pointswith stabilizer of dimension n . Hence, neither does Bl π X max X . This proves (3). (cid:3) Functoriality for strong morphisms of singular stacks.
Theorem 6.5 can beextended to strong morphisms of singular stacks with stable good moduli spaces. Let f : Y → X be a strong morphism of stacks with stable good moduli space morphisms π X : X → X , π Y : Y → Y . Assume that every point of both X s and Y s has stabilizer ofa fixed dimension n ; e.g., that X and Y are connected or properly stable. Let X ′ → X and Y ′ → Y be the canonical morphisms produced by Theorem 2.11. Proposition 8.5 (Functoriality for strong morphisms) . Under the assumptions above, Y ′ is the schematic closure of Y s in Y × X X ′ .Proof. Under our assumptions Y ′ and X ′ are produced by taking repeated saturatedblowups in X max and Y max respectively. If f − ( X max ) = ∅ , then Y × X Bl X max X = Y .If f − ( X max ) = ∅ , then the maximal stabilizer dimensions of X and Y coincide and Y max = f − ( X max ) since f is strong. Hence by Proposition 3.11, the saturated blowupBl π Y Y max Y → Y is the the strict transform of Bl π X X max X → X in the sense of Definition3.10; that is, it is the schematic closure of Y r Y max in Y × X Bl π X X max X . After the finalblowup, Y s becomes schematically dense in Y ′ and the result follows. (cid:3) Example 8.6.
The following example shows that if Y is singular, then Y ′ need notequal the fiber product Y × X X ′ even if the morphism Y → X is lci.Let X = [ A / G m ] where G m acts by weights (1 , , − X max is the origin.Let Y ⊂ X be the closed substack defined by the ideal ( xz ). Since xz is invariant, f : Y → X is a strong regular embedding. The induced morphism of good modulispaces is the closed immersion A → A .Since the maximum dimension of a stabilizer is one, the canonical reduction ofstabilizers is obtained by a single saturated blowup along X max . Since X is smooth,this is also the Reichstein transform. The saturation of X max is the substack definedby the ideal ( xz, yz ) which is the union of two irreducible substacks: the divisor V ( z )and the codimension-two substack V ( x, y ). We conclude that X ′ = (cid:2) Spec k [ xz , yz , z ] r V ( xz , yz ) / G m (cid:3) . and thus X ′ × X Y ⊂ X ′ is the closed substack defined by the ideal ( z xz ).On the other hand, the strict transform of Y is Y ′ = Bl π Y Y max Y which is cut out by ( xz ). Remark 8.7 (Symmetric and intrinsic blowups) . Let X be a stack with a properlystable good moduli space. Then our algorithm repeatedly makes saturated blowupsin X max . An alternative would be to replace the saturated blowup by the saturated symmetric blowup : if X max is defined by the ideal sheaf J X , we may take the saturatedProj of the symmetric algebra Sym( J X ) instead of the saturated blowup which is thesaturated Proj of the Rees algebra L k ≥ J k X . When X is smooth, then so is X max andthe symmetric algebra of J X coincides with the Rees algebra of J X . If X → Y is aclosed embedding into a smooth stack with a good moduli space, then L k ≥ J k Y =Sym( J Y ) → Sym( J Y ⊗ O Y O X ) → Sym( J X ) → L k ≥ J k X are surjective. We thus haveclosed embeddingsBl X max X →
SymBl X max X → (Bl Y max Y ) × Y X → Bl Y max Y . In particular, the maximum stabilizer dimension of SymBl X max X also drops.For a quasi-projective scheme U over C with an action of a reductive group G ,Kiem, Li and Savvas [KL13, KLS17] have defined the intrinsic blowup Bl G U of U . If U → V is a G -equivariant embedding into a smooth scheme, then we have an inducedembedding of stacks X = [ U ss /G ] → Y = [ V ss /G ] that admit good moduli spaces. Thestack [Bl G U/G ] is a closed substack of (Bl Y max Y ) × Y X that is slightly larger than boththe blowup and the symmetric blowup of X in X max and is independent on the choice ANONICAL REDUCTION OF STABILIZERS OF ARTIN STACKS 29 of a smooth embedding. It would be interesting to find a definition of the intrinsicblowup for a stack with a good moduli space that (1) does not use smooth embeddingsand (2) does not use a presentation X = [ U ss /G ]. The intrinsic blowup also seems tobe related to derived stacks and blowups of such and it would be interesting to describethis relationship. Appendix A. Gerbes and good moduli spaces
Let π : X → X be a good moduli space morphism. In this appendix, we study when π factors as X → X tame → X where X → X tame is a gerbe and X tame → X is a coarsemoduli space. A necessary condition is that the stabilizers of X have locally constantdimension. We prove that this is sufficient when X is reduced (Proposition A.2) andthat X tame is a tame stack. When X is not reduced, we give a precise condition inCorollary B.7. When there is a factorization X → X tame → X as above, then X → X isa homeomorphism and in fact a coarse moduli space in the sense of [Ryd13, Definition6.8]. Proposition A.1.
Let π : X → X be the good moduli space of a stack such that allstabilizers are 0-dimensional. Then X is a tame stack and X is also the coarse modulispace of X . Moreover, X is separated if and only X is separated.Proof. By assumption, X has quasi-finite and separated diagonal (recall that our stackshave affine, hence separated, diagonals). Since X has a good moduli space, it followsthat X has finite inertia [Alp14, Theorem 8.3.2], that is, X is tame and X is its coarsemoduli space. Moreover, π is a proper universal homeomorphism, so X is separated ifand only if X is separated [Con05, Theorem 1.1(2)]. (cid:3) We can generalize the previous proposition to stacks with constant dimensionalstabilizers.
Proposition A.2.
Let X be a reduced Artin stack with good moduli space π : X → X .If the dimension of the stabilizers of points of X is constant, then X is a gerbe over atame stack X tame whose coarse space is X . In particular, if X is smooth, then X tame issmooth and X has tame quotient singularities. A.1.
Reduced identity components.
To prove the proposition, we need some pre-liminary results on reduced identity components of group schemes in positive charac-teristic.Let G be an algebraic group of dimension n over a perfect field k . By [SGA3, Expos´eVIa, Proposition 2.3.1] or [Sta16, Tag 0B7R] the identity component G of G is an openand closed characteristic subgroup. Let G = ( G ) red (non-standard notation). Sincethe field is perfect, G is a closed, smooth, subgroup scheme of G [SGA3, Expos´e VIa,0.2] or [Sta16, Tag 047R]. Moreover, dim G = dim G = n . Remark A.3.
In general, G is not normal in G , for example, take G = G m ⋉ ααα p . Butif G is diagonalizable, then G ⊂ G is characteristic, hence G ⊂ G is normal. Indeed, this follows from Cartier duality, since the torsion subgroup of an abelian group is acharacteristic subgroup. Lemma A.4.
Let S be a scheme and let G → S be a group scheme of finite typesuch that s dim G s is locally constant. Let H ⊂ G be a subgroup scheme such that H s = G s, for every geometric point s : Spec K → S . If S is reduced, then there is atmost one such H and H → S is smooth.Proof. If S is reduced, then H → S is smooth [SGA3, Expos´e VIb, Corollaire 4.4].If H and H are two different subgroups as in the lemma, then so is H ∩ H . Inparticular, H ∩ H is also smooth. By the fiberwise criterion of flatness, it follows that H ∩ H = H = H . (cid:3) Note that the lemma is also valid if S is a reduced algebraic stack by passing to asmooth presentation. Definition A.5. If S is reduced and there exists a subgroup H ⊂ G as in the lemma,then we say that H is the reduced identity component of G and denote it by G . Proposition A.6.
Let X be a reduced algebraic stack such that every stabilizer hasdimension d . If either (1) char k = 0 , or (2) X admits a good moduli space,then there exists a unique normal subgroup ( I X ) ⊂ I X such that I X → X is smoothwith connected fibers of dimension d . Moreover, when X admits a good moduli space,then ( I X ) ⊂ I X is closed and I X / ( I X ) → X is finite.Proof. If char k = 0, then the fibers of I X → X are smooth. It follows that the identitycomponent ( I X ) is represented by an open subgroup which is smooth over X [SGA3,Expos´e VIb, Corollaire 4.4]. The identity component is always a normal subgroup.If X instead admits a good moduli space, then we proceed as follows. By the localstructure theorem of [AHR15, Theorem 2.9], for any closed point x ∈ X ( k ) there is anaffine scheme U = Spec A and a cartesian diagram of stacks and good moduli spaces[Spec A/G ] / / (cid:15) (cid:15) X π (cid:15) (cid:15) Spec( A G ) / / X where the horizontal maps are ´etale neighborhoods of x and π ( x ) respectively and G = G x is the stabilizer at x . Since, the diagram is cartesian, the map [ U/G ] → X isstabilizer preserving so the diagram of inertia groups I ([ U/G ]) / / (cid:15) (cid:15) I X (cid:15) (cid:15) [ U/G ] / / X ANONICAL REDUCTION OF STABILIZERS OF ARTIN STACKS 31 is also cartesian. Since [
U/G ] is a quotient stack I ([ U/G ]) = [ I G U/G ] where I G U = { ( g, u ) : gu = u } ⊂ G × U is the relative inertia group for the action of G on U . Here G acts on I G U via h ( g, u ) = ( hgh − , hu ).Note that [ G × U/G ], I ([ U/G ]) ⊂ [ G × U/G ] are group schemes over [
U/G ] withfibers of dimension d , and that [ G × U/G ] → [ U/G ] is smooth (a twisted form of G ).It follows that I ([ U/G ]) exists and equals [ G × U/G ].By Nagata’s theorem, G is diagonalizable [DG70, IV, §
3, Theorem 3.6]. Thus G ⊂ G is normal (Remark A.3), and hence so is I ([ U/G ]) ⊂ I ([ U/G ]).Since ( − ) is unique and commutes with ´etale base change, it follows by descentthat ( I X ) exists and is a normal closed subgroup.Finally, we note that I X / ( I X ) is finite since I G U/ ( G × U ) ⊂ ( G/G ) × U is aclosed subgroup of a finite group scheme. (cid:3) Note that we in the proof worked with the reduced stack [
U/G ] rather than withthe scheme U which perhaps is not reduced. If X is smooth, then one can arrange that U is smooth [AHR15, Theorem 1.1]. Proof of Proposition A.2.
We have seen that the inertia stack I X → X contains aclosed, normal subgroup I X which is smooth over X , such that I X /I X → X is finitewith fibers that are linearly reductive finite groups (Proposition A.6). By [AOV08,Appendix A], X is a gerbe over a stack X ((( I X which is the rigidification of X obtainedby removing I X from the inertia. The stack X tame = X ((( I X will be the desiredtame stack. In the ´etale chart in the proof of Proposition A.6, we have that X tame =[ U/ ( G/G )].The inertia of X tame is finite and linearly reductive because its pull-back to X co-incides with I X /I X (or use the local description). Moreover, X → X tame has theuniversal property that a morphism
X → Y factors (uniquely) through X tame if andonly if I X → I Y factors via the unit section Y → I Y . In particular we obtain afactorization X → X tame → X and X tame → X is the coarse moduli space since it isinitial among maps to algebraic spaces. (cid:3) Remark A.7. If X is not reduced, then it need not be a gerbe over a tame stack. Forexample, take X = [Spec k [ x ] / ( x n ) / G m ] where G m acts by multiplication. Remark A.8. If X is as in Proposition A.6 and char k = p , then in general there isno subgroup ( I X ) . For a counter-example, take X = BG with G = G m ⋉ ααα p . Then I ( BG ) is a reduced algebraic stack. Also, even if there is an open and closed subgroup( I X ) with connected fibers, this subgroup need not be flat. For a counter-example,take X = [Spec k [ x ] /µµµ p ] where µµµ p acts with weight 1. Then I X has connected fibersbut is not flat. Appendix B. Fixed loci of Artin stacks
Lemma B.1.
Let X be an Artin stack. Then the locus of points with maximal-dimensionalstabilizer is a closed subset of |X | . Proof.
Since the representable morphism I X → X makes I X into an X -group, thedimension of the fibers of the morphism is an upper semi-continuous function on |X | .Thus the locus of points with maximal-dimensional stabilizer is closed. (cid:3) If a group scheme
G/k acts on X , then the fixed locus X G can be given a canonicalscheme structure: X G represents the functor of G -equivariant maps T → X where T is equipped with the trivial action. The functor X G is represented by a closed sub-scheme [CGP10, Proposition A.8.10 (1)] because the coordinate ring of G is a projective(even free) k -module. Moreover, if G is linearly reductive and X is smooth, then X G is also smooth [CGP10, Proposition A.8.10 (2)] Proposition B.2. If X is smooth and admits a good moduli space, then the locusof points X n with stabilizer of a fixed dimension n (with its reduced induced substackstructure) is a locally closed smooth substack. In particular, the locus X max of pointswith maximal-dimensional stabilizer is a closed smooth substack. Moreover, if E is ansnc divisor, then X n meets E with normal crossings.Proof. We may replace X with its open substack where every stabilizer has dimensionat most n . Let x be a closed point of X n = X max and let G x be its stabilizer group.By the local structure theorem [AHR15, Theorem 2.9, Theorem 1.1] there is a smooth,affine scheme U = Spec A with an action of G x and a cartesian diagram of stacks andmoduli spaces [ U/G x ] / / (cid:15) (cid:15) X π (cid:15) (cid:15) U//G x / / X (cid:3) where the horizontal arrows are ´etale and u ∈ U is a fixed point above x . It follows that[ U/G x ] n is the inverse image of X n under an ´etale morphism. In particular [ U/G x ] n (with its reduced induced stack structure) is smooth at u if and only if X n is smoothat x .As in § A.1, let G be the reduced identity component of G x . Then dim G = dim G = n and any n -dimensional subgroup of G necessarily contains G .Since G/G is finite, hence affine, so is BG → BG . It follows that BG is cohomo-logically affine, that is, G is linearly reductive.Since n = dim G x , a point of U has stabilizer dimension n if and only if it isfixed by the linearly reductive subgroup G . Thus [ U/G x ] n = [ U G /G x ]. By [CGP10,Proposition A.8.10(2)] U G is also smooth. Note that G x acts on U G because G is acharacteristic, hence normal, subgroup of G x (Remark A.3).If E = P ni =1 E i is an snc divisor on X , let D = P ni =1 D i denote the pull-back to U .For I ⊂ { , , . . . , n } , let D I = ∩ i ∈ I D i . First note that D is snc at u . Indeed, since BG x → [ D I /G x ] is a regular embedding, so is u → D I , for every I . Since D G I = U G ∩ D I is smooth at u [CGP10, Proposition A.8.10(2)], it follows that U G meets D with normal crossings at u . By flat descent, X n meets E with normal crossings. (cid:3) ANONICAL REDUCTION OF STABILIZERS OF ARTIN STACKS 33
Remark B.3.
Proposition B.2 holds more generally for any smooth algebraic stack X such that the stabilizer of every closed point has linearly reductive identity component.Indeed, for any such point x there is an ´etale morphism [ U/G x ] → X and [ U/G x ] n =[ U G /G x ] is smooth.If X is an arbitrary algebraic space and X = [ X/G ], then we can equip X max withthe substack structure of [ X G /G ]. The next proposition shows that this substackstructure on X max is independent on the presentation X = [ X/G ]. Combining this factwith the local structure theorem we can conclude that if X is an arbitrary stack witha good moduli space, then X max has a canonical scheme structure. To achieve this, westart with a slightly different definition of X max , not referring to G . Proposition B.4.
Let X be an Artin stack that admits a good moduli space. Let n be themaximal dimension of the stabilizer groups. Consider the functor F : (Sch / X ) op → (Set) where F ( T → X ) is the set of closed subgroups H ⊂ I X × X T that are smooth over T with connected fibers of dimension n . Then the functor is represented by a closedsubstack X max = X n . In particular, for any T → X , there is at most one such subgroup H and it is characteristic. Moreover, (1) |X max | is the set of points with stabilizer of dimension n . (2) If X = [ X/G ] where G is linearly reductive, then X n = [ X G /G ] . (3) The following are equivalent (a) X = X max , (b) I X contains a closed subgroup,smooth over X with connected fibers of dimension n , and (c) I X contains aclosed subgroup, flat over X with fibers of dimension n . (4) When X is smooth, X max is smooth. Note that from the functorial description of X max , it follows that if X ′ → X is astabilizer-preserving morphism, then X ′ max = X max × X X ′ . Proof of Proposition B.4. (1) is an immediate consequence of the main claim sinceif T = Spec k with k perfect, then H = ( I X × X T ) is the unique choice of H .(3a) = ⇒ (3b) follows by definition and (3b) = ⇒ (3c) is trivial.To prove the main claim and (3c) = ⇒ (3a) we may work fppf-locally around a pointwith stabilizer of dimension n . Using the local structure theorem of [AHR15], we canthus assume that X = [ X/G ] where G is linearly reductive of dimension n . The mainclaim thus follows from (2). The following lemma applied to H = I X × X T implies (2)and (3c) = ⇒ (3a). If X is smooth, then X G is smooth [CGP10, Proposition A.8.10(2)] and (4) follows. Lemma B.5.
Let G be a linearly reductive group scheme of dimension n over k . Let T be a scheme and let H ⊂ G × T be a closed subgroup. Then the following are equivalent: (1) G × T ⊂ H . (2) There exists a closed subgroup H ⊂ H that is smooth over T with connectedfibers of dimension n . (3) There exists a closed subgroup H ⊂ H that is flat over T with fibers of dimen-sion n . and under these conditions H is unique and equals G × T .In particular, the functor F representing closed subgroups H ⊂ H that are smoothwith connected fibers of dimension n coincides with the functor representing that G × T ⊂ H and this is represented by a closed subscheme T n of T .Proof. We can replace G with G and H with H ∩ G × T and thus assume that G isconnected. We can also assume that T is affine.If char k = 0, then G is reduced and connected so any closed subgroup H ⊂ G × T that is flat with fibers of dimension n necessarily equals G × T by the fiberwise criterionof flatness.If char k = p , then G is diagonalizable by Nagata’s theorem [DG70, IV, §
3, Theorem3.6] so G × T is the unique closed subgroup of G × T that is smooth of dimension n .Indeed, we have seen uniqueness when T is reduced (Lemma A.4) and the uniquenessin general follows from rigidity of groups of multiplicative type [SGA3, Expos´e X,Corollaire 2.3, Expos´e IX, Corollaire 3.4bis].Thus, the functor F represents when G × T ⊂ H , or equivalently, when H ∩ ( G × T ) ⊂ G × T is the identity, that is, the Weil restriction F = Y G × T/T ( H ∩ G × T ) / ( G × T ) . This functor is represented by a closed subscheme since G × T → T is pure/essentiallyfree, see [SGA3, Expos´e VIII, Th´eor`eme 6.4], [RG71, Th´eor`eme 4.1.1] or [CGP10,Proposition A.8.10 (1)]. (cid:3) Remark B.6.
Similarly, we have canonical locally closed substacks X n for any integer n ≥ Corollary B.7.
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Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri65211
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