FUNCTORIAL RESOLUTION OF TAME QUOTIENT SINGULARITIES IN POSITIVE CHARACTERISTIC. Contents

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1 FUNCTORIAL RESOLUTION OF TAME QUOTIENT SINGULARITIES IN POSITIVE CHARACTERISTIC FEDERICO BUONERBA Abstract. The object of the present is a proof of the existence of functorial resolution of tame quotient singularities for quasi-projective varieties over algebraically closed fields. Contents I. Generalities 4 I.I. Tame group actions 4 I.II. Stacks with quotient singularities 5 I.III. Actions of µ l 9 I.IV. Weighted blow-ups 10 I.V. Toric varieties 12 II. Resolution 14 II.I. Step 1: Reduction to tame cyclic quotient singularities. 14 II.II. Step 2: Resolution of diagonal cyclic quotient singularities. 16 II.III. Final assembly 18 References 18 The role of quotient singularities, in the general problem of the resolution of singularities in positive characteristic, has been highlighted in the fundamental work of de Jong [dj96], where the general problem has been reduced to the more specific one of understanding singularities created by inseparable morphisms and group actions. Theorem. [dj96, 7.4] Let X be a projective variety over an algebraically closed field. There exists a radicial morphism Y X and a modification Z Y such that Z has at worst quotient singularities. 1

2 2 FEDERICO BUONERBA A huge class of group actions enjoy the property of being linearizable, i.e. the action is formally equivalent to that of a group of linear endomorphisms of vector space. In this situation the singularities created by the action are easier to handle, since the existence of formal coordinates along which the action is linear provides a rich amount of information. By way of examples, all tame abelian group actions, see I.I, create singularities that live, étale-locally, in the world of toric varieties, see I.V.1, and their resolution is in fact a key step for those positive characteristic oriented constructions that lead to a resolution of singularities in characteristic zero, such as those appearing in [BP96],[dJ96]. As well known, étale-locally the resolution problem for tame abelian quotient singularities is completely understood, i.e. there are explicit algorithms, such as [KKMS73, theorems 11 and 11*], and [MP13, III.iii.4.bis], that provide us with a resolution. The issue is therefore the local-toglobal step, where the toric machinery is not sufficiently strong without additional global assumptions, for example the existence of a toroidal embedding. Consequently it seems convenient to look for a resolution procedure that is functorial under smooth localization, as introduced in [Wlo05, 1.0.3]. Our main result is: Main Theorem. Let k be an algebraically closed field. Let Q k denote the category of quasi-projective varieties with tame quotient singularities, and Sm k that of smooth quasiprojective varieties. Then there exists a resolution functor R : Q k Sm k and, for any X Ob(Q k ), a birational and projective morphism r X : R(X) X, which is an isomorphism over the smooth locus of X. The resolution functor commutes with smooth localization, that is if f : Y X is a smooth arrow in Q k, there exists a unique Y -isomorphism φ f : f R(X) R(Y ). Now we can discuss the ideas employed. Following the general pattern, [Kol07], [Wlo05], that the construction of a resolution functor is algorithmic in its own nature, we define an invariant X i(x) N, such that i(x) = 0 if and only if X is smooth. Then we proceed by induction on i, by performing birational operations - smooth and weighted blow-ups along smooth centers - that eventually decrease i. The proof consists of two steps: the first step II.I constructs a birational modification whose resulting space has only tame cyclic quotient singularities, and the invariant does not increase. Here the full power of the philosophy functoriality in the smooth topology appears, and most of the logical difficulties, apart from some routine technical issues, are hidden in the inductive step of

3 FUNCTORIAL RESOLUTION OF TAME QUOTIENT SINGULARITIES 3 the process. This inductive step resolves the singularities of a Deligne-Mumford stack with smaller invariant, and the variety we are looking for is its GC quotient, whose existence as an algebraic space is granted by [KM97]. At this point the inductive hypotheses show up again and force the GC quotient to be a quasi-projective variety, if we started with such. The second step II.II is an algorithm that resolves the singularities, if we start with tame cyclic quotient singularities. It is typical of this sort of problems, [KKMS73], that the local resolution is pretty straightforward, and indeed there are several choices - coordinates, generators of cyclic groups, etc. etc. - that can simplify the process even more. On the other hand, the number of local choices is directly proportional to the difficulty in globalizing the local construction, and it is doubtless that we need some extra data to overcome this issue. In our tame cyclic quotient case, it is enough to have an equivariant divisor, on the Vistoli covering stack, along which the stabilizers act with a faithful character. The logic how the two steps fit together and provide a resolution is as follows: starting with a variety with tame quotient singularities, the first step produces a variety with tame cyclic quotient singularities and a marked irreducible divisor, II.I.5. Properties of this pair are such that we can employ it as input for our second step, whose output is a smooth variety. Functoriality in the smooth topology implies that the resolution procedure applies to Artin stacks, II.I.2. It is worth remarking that, in the course of the proof, we will encounter singularities created by diagonal actions of group schemes of roots of unity, which are not necessarily tame. This must happen if one wants to tackle the problem using weighted modifications/toric geometry in positive characteristic. It turns out that, in this specific circumstance, Mumford s resolution process for toroidal singularities can be carried over by weakening a bit the assumption on the existence of a toroidal embedding, whose role will be played by a divisor along which cyclic stabilizers act with faithful character, as in II.II.2. Unfortunately, the methods developed here seem to completely lose their efficiency if one drops the tameness assumption, in fact the crucial fact we use profusely is that there exist regular parameters along which tame abelian actions are diagonalized, while of course this never happens with non-trivial p-group actions in characteristic p. We suspect that a resolution functor that deals with all possible quotient singularities simultaneously must proceed via a completely different strategy. The techniques developed here work in a much larger generality. The details required to

4 4 FEDERICO BUONERBA deal with excellent noetherian schemes with tame quotient singularities will appear in a subsequent paper. Acknowledgments: This problem has been suggested by Fedor Bogomolov. His encouragement, along with several clarifications, have been of great help. This work could never be completed without the support of Michael McQuillan, whose restless and careful explanations permeate and shape the core ideas building our proof. Grateful thanks are extended to Gabriele Di Cerbo, for his editing that drastically improved our presentation. It is a pleasure to acknowledge the comments of the referee, for fundamental suggestions correcting several mistakes and improving the exposition. Finally, Daniel Bergh has suggested connections with his work [Be14]. I. Generalities I.I. Tame group actions. Let k be an algebraically closed field and G be a finite group acting by k-automorphism on a noetherian, regular, local k-algebra O. The action is called tame if ( G, char(k)) = 1. If G acts on a smooth algebraic variety by k-automorphisms, we say that the action is tame if it is generically free, and the stabilizer at any geometric point acts tamely on the corresponding local ring. From the point of view of singularities we could request more, i.e. the action to be free in codimension one, by way of the celebrated Chevalley-Shephard-Todd theorem: Fact I.I.1. If G is a finite group acting tamely on a smooth algebraic variety, then for any geometric point x with stabilizer G x, the ring of invariants Ox Gx ring if and only if G x is generated by pseudoreflections. O x is a regular local Tame actions enjoy some remarkable properties, with respect to the local algebra they are the simplest possible. We can recollect those properties that we need in the following simple propositions: Proposition I.I.2. Let G be a cyclic group, generated by g G, acting tamely on a noetherian, regular, local k-algebra O with maximal ideal m. Then there are elements ζ 1,..., ζ n k and a regular system of parameters x 1,..., x n generating m such that x g i = ζ i x i. Proof. As g acts on the k-vector space m/m 2, by assumption we can certainly find parameters and eigenvalues satisfying the identities x g i = ζ i x i mod m 2. We can replace each x i by

5 y i = G 1 k= G 1 ζi k, and observe that y i = x i mod m 2, and y i satisfies the requi- red relation. FUNCTORIAL RESOLUTION OF TAME QUOTIENT SINGULARITIES 5 k=0 x gk i Corollary I.I.3. If G acts tamely on a smooth algebraic variety X then the subvariety X G of points fixed by G is smooth. Proof. Let O be the local ring at a geometric point x fixed by G, then (essentially) as in the previous proof we can average any isomorphism Ô k[[x 1,..., x n ]] in order to make it equivariant with respect to the induced action of G on the power series ring. Here the action is by linear automorphism, whence the ideal defining X G is easily seen to be generated by the linear forms (x g i x i) g G, and consequently the subvariety X G is smooth at x. Proposition I.I.4. Let O be a strictly henselian regular local k-algebra with maximal ideal m, G a finite group acting tamely on it. If I is a principal ideal which is fixed by G then there exists a character χ : G k and a generator f of I, such that f g = χ(g)f for any g G. Proof. Let f be a generator of I. By assumption, for every g there is u g O such that f g = u g f. It follows that f gh = u h gu h f, whence the set map G O given by g u g is a 1-cocycle representing a class in H 1 (G, O ). Consider the exact sequence 0 (1 + m) O k 0 of multiplicative groups. By assumption on O being strictly henselian, since G is coprime to char(k) we see that 1 + m is G -divisible, whence H 1 (G, 1 + m) = H 2 (G, 1 + m) = 0. Consequently we have a natural isomorphism H 1 (G, O ) H 1 (G, k ). Moreover since the G-action on the residue field k is trivial, we get H 1 (G, k ) = Hom(G, k ), whence, upon replacing f by uf, for some invertible u, the map g u g is a character of the group G, and of course uf is the required generator. Proposition I.I.5. Let q : O O be an injective morphism of rings, equivariant by the action of a cyclic group C. Assume C acts with faithful character on f O, then C acts with faithful character on q(f). Proof. Let s be the minimal integer such that χ s q(f) = q(f), then q(χ s f f) = 0, and since q is injective we deduce s = C, that is the action is faithful. I.II. Stacks with quotient singularities. Let k be an algebraically closed field. algebraic variety X/k has quotient singularities if it is normal and for any geometric point An

6 6 FEDERICO BUONERBA x, there exist a finite group G x acting on a smooth complete local k-algebra k[[x 1,..., x n ]] and an inclusion i x : Ô x k[[x 1,..., x n ]] such that i x (Ôx) = k[[x 1,..., x n ]] Gx. By Artin s approximation this is equivalent to the fact that X admits an étale cover V i X, where V i = U i /G i for smooth varieties U i and finite groups G i, whose action on U i is generically free. X has tame quotient singularities if each G i acts on U i tamely. Similarly, an Artin stack X /k has quotient singularities if for some, hence every, smooth atlas V X, the algebraic variety V has quotient singularities. In the sequel we will use the following notation: Definition I.II.1. Let X be an algebraic variety with tame quotient singularities. The minimal G x such that X is, étale-locally at x, of the form U x /G x for some smooth U x, will be called the geometric stabilizer at x. For an Artin stack X with smooth atlas V X, the geometric stabilizer at a geometric point x is the geometric stabilizer at the corresponding geometric point x V. Since tame group actions are formally linear, the notion of geometric stabilizer is well defined by [Kol07, ]. Before getting into the properties of Deligne-Mumford stack with tame quotient singularities, we recall an extremely useful local description of Deligne-Mumford stacks: Fact I.II.2. [AV01, 2.2.3] Let X be a Deligne-Mumford stack of finite type over a field, then it admits an étale cover by classifying stacks, i.e. stacks of the form [Y/G] with G a finite group. In the situation where X is a Deligne-Mumford stack with tame quotient singularities there exists a canonical, smooth cover of X, which we will refer to as Vistoli covering stack. The construction appears in the proof of [Vis89, 2.8], and we reproduce it for convenience of the reader. Fact I.II.3. [Vis89, 2.8] Let X be a Deligne-Mumford stack with tame quotient singularities. There exists a smooth Deligne-Mumford stack X V with a morphism X V X which is étale in codimension one, satisfying the following universal property: if Y is a smooth Deligne-Mumford stack and Y X is étale in codimension one, then there exists a unique factorization Y X V X. Proof. Let W X be an étale atlas with tame quotient singularities, say W = U i /G i and let (s, t) : R X = W X W W. Denote by H i the normal subgroup of G i generated

7 FUNCTORIAL RESOLUTION OF TAME QUOTIENT SINGULARITIES 7 by pseudo-reflections, let W V = U i /H i and R V be the normalization of W V X W V U i /H i s (U i /H i ) U i /H i X U j /H j s U i /G i R X t (U j /H j ) t normalization R V X U j /G j U j /H j R V is a normal algebraic variety since the diagonal of X is representable. The two projections (s V, t V ) : R V W V are finite and étale in codimension one, since the action of G i /H i on U i /H i is free in codimension one for every i. W V is smooth by I.I.1, whence the Zariski- Nagata purity theorem [Gro68, X.3.4] implies that the projections s V, t V are everywhere étale and the groupoid X V = [W V /R V ] is a smooth Deligne-Mumford stack satisfying the required properties. Remark I.II.4. The Vistoli cover X V X is, locally on an étale atlas around a geometric point x, given by the action of the geometric stabilizer U x U x /G x in the notation I.II.1. Similarly in the situation where X is a normal DM stack and D is a Q-Cartier divisor, such that for any geometric point x X, the minimal integer number n(x) such that n(x)d x is Cartier satisfies (n(x), char(k)) = 1, there is a canonical cyclic cover of X in which D becomes everywhere Cartier. We will denote this stack by X (D), the Cartification of D. The construction is analogous to that of the Gorenstein covering stack, and again, for the convenience of the reader, it is described below: Proposition I.II.5. [McQ05, I.5.3] There exists a cyclic cover q D : X (D) X such that qd D is Cartier, satisfying the following universal property: if Y is a Deligne-Mumford stack and q : Y X is such that q D is Cartier, then there exists a unique factorization Y X (D) X. Proof. Let x be a geometric point in X, V an étale neighborhood and V o its smooth locus. Up to shrinking, we can assume that nd is generated, over V, by a local section f and that D is generated, over V o, by a local section t. Inside the geometric line bundle Spec (Sym O V o(d) ) V o consider the subvariety W o defined by the ideal T n f =

8 8 FEDERICO BUONERBA 0, where T is a generator of Sym O V o(d) as an O V o-algebra. This is a cover of V o corresponding to the extraction of an n-th root of the unity u OV o satisfying tn uf = 0, which is étale since (n, char(k)) = 1. This cover extends canonically to a ramified cover V (D) V by taking the integral closure of O V in the function field of W o. This gives a local description, with respect to an atlas of X, of an atlas for X (D), in particular the étale local ring of X (D) at a geometric point x is O x [T ]/(T n(x) f). To deduce the groupoid relation, we proceed as in the Vistoli situation, thus we need to check that the natural morphism (V (D) X V (D)) norm V (D) is étale. After étale localization around a geometric point, this amounts to proving that the natural morphisms O[T ]/(T n f) O[T, S]/(T n f, S m g) norm O[T ]/(S m g) O[T, S]/(T n f, S m g) norm are étale, where O is a strictly henselian local ring with a height one prime ideal I such that (f) = I n and (g) = I m. Let d = gcd(n, m), n = n d, m = m d and w = nm/d. There is a unit u O such that f m = ug n, thus we have a relation T w = us w, and since O is strictly henselian and (w, char(k)) = 1 we see that u 1/w O. Consequently we deduce a relation ζ w =1 T ζu1/w S = 0 in O[T, S]/(T n f, S m g). Taking its normalization we get a product of rings O[T, S]/(T n f, S m g, T ζu 1/w S) O[T ]/(T n f) ζ w =1 ζ w =1 which implies that the natural morphisms defining the Cartification groupoid O[T ]/(T n f) O[T ]/(T n f) ζ w =1 O[S]/(S m g) O[T ]/(T n f) ζ w =1 given respectively by T ζ w =1 T and S ζ w =1 ζu1/w T are indeed étale, thus proving the existence of X (D) as a Deligne-Mumford stack. The universal property follows by the local nature of the construction. Remark I.II.6. The construction of the Cartification morphism depends heavily on the assumption that the Cartier index of the divisor is coprime to the characteristic of the base field. Without this assumption the local Cartification factors through a non trivial inseparable quotient, and the proposition is false.

9 FUNCTORIAL RESOLUTION OF TAME QUOTIENT SINGULARITIES 9 In the sequel we will need a reformulation of the universal property of the Cartification. Lemma I.II.7. Cartification is stable under flat pullbacks, that is if f : Y X is a flat morphism and D is Q-Cartier on X then there exists a canonical isomorphism i f : Y (f D) f X (D). Proof. This is an easy application of the universal property: D becomes Cartier on Y (f D) and this affords a morphism i f : Y (f D) f X (D). Since D is Cartier on X (D), so it is after pullback to f X (D), whence by the universal property we deduce an inverse to i f. I.III. Actions of µ l. In this section we fix some notation regarding actions of diagonalizable group schemes. Recall, by definition, µ l := Spec Z[T ]/(T l 1). Definition I.III.1. Let O be a regular local k-algebra of dimension n, and l > 0 any natural number. A µ l -action on O with characters (a 1,..., a n ) (Z/lZ) n, is a morphism O O[T ]/(T l 1) such that there exists a regular system of parameters x 1,..., x n with x i T a i x i. The equalizer of this action will be denoted by O µ l. Similarly, we can define a µ l -action on a smooth Artin stack X and, upon fixing a local generator of µ l, it is clear what it means for the action to have characters (a 1,..., a n ) around a fixed geometric point. Obviously, as in section I.II, we have: Definition I.III.2. An algebraic variety X has diagonal cyclic quotient singularities if, for every geometric point x X, there exists a strictly henselian regular local ring Ox V with a µ l,x -action, such that O x = (Ox V ) µ l,x. Similarly, an Artin stack X has diagonal cyclic quotient singularities if some smooth atlas does. There is a subtle choice in the above definitions, which is that of a generator, T, of µ l. Of course, any other choice, say an isomorphism O[T ]/(T l 1) O[S]/(S l 1) given by T S d, with (d, l) = 1, will lead to the same equalizer, but the characters of the action will obviously change. The important point to keep in mind is: Remark I.III.3. There is no canonical way to attach characters, as in definition I.III.1, to a local ring with diagonal cyclic quotient singularities. Finally, we need the notion of group scheme acting faithfully across a divisor:

10 10 FEDERICO BUONERBA Definition I.III.4. (i) Let O be a regular strictly henselian local ring with a µ l -action. Let f O be a non-unit, then we say that the action is faithful across the divisor D(f) = (f = 0) if f T f in the notation of definition I.III.1. (ii) Let O be a strictly henselian local ring with diagonal cyclic quotient singularities, and D a Q-Cartier divisor. We say that the stabilizer acts faithfully across D if there exists O V such that (i) holds for the Cartier divisor D V in O V. In other words, (ii) means that stabilizers act faithfully across a divisor if this is true, étale-locally, on some smooth cyclic cover. I.IV. Weighted blow-ups. Let X be a smooth Deligne-Mumford stack, and (a 1,..., a r ) an r-tuple of natural numbers. A blow-up with weights (a 1,..., a r ) is the projectivization Proj X ( k 0 I k /I k+1 ) X where I k is a sheaf of ideals such that, étale locally, there exist functions x 1,..., x r forming part of a regular system of parameters and I k is generated by monomials x α xαr r with a 1 α a r α r k. Then k 0 I k /I k+1 is a sheaf of finitely generated graded algebras. To give a local description, assume we are blowing up the origin in the affine space over k with weights (a 1,..., a n ). It is covered by affine open sets V i = Spec(R i ) where R i = k[ xα xαn n, a i c i = a 1 α a n α n ] x c i i and the morphism R i k[y 1,..., y n ] given by x i y a i i, x j y j y a j i induces an isomorphism Spec(k[y 1,..., y n ] µa i ) = A n k /µ ai Spec(Ri ) where the the action is expressed, in terms of a generator of µ ai, by y i T y i, y j T a j y j. In other words, V i = Spec(R i ) where R i is the algebra obtained as equalizer of the µ ai - action on k[y 1,..., y n ] with characters ( a 1,..., a i 1, 1, a i+1,..., a n ). Finally, the exceptional divisor has equation y i = 0 in the i-th chart. We will be interested in weighted blow-ups induced by µ l -actions: Definition I.IV.1. Notation as in I.III.1, the weighted algebra induced by the characters (a 1,..., a n ) is the sheaf of algebras A(a 1,..., a n ) = k 0 I k /I k+1 on Spec(O), where I k := (x α xαn n a 1 α a n α n k)

11 FUNCTORIAL RESOLUTION OF TAME QUOTIENT SINGULARITIES 11 µ l acts naturally on A(a 1,..., a n ) and its fixed sub-algebra B(a 1,..., a n ) := A(a 1,..., a n ) µ l is canonically a sheaf of algebras on Spec(O µ l). In light of remark I.III.3, it is natural, and rather important, to ask: given, say, a strictly henselian local ring O with diagonal cyclic quotient singularities, and a regular cover O V with µ l -action, what is the dependence of the filtration I k, as in definition I.IV.1, on the choice of characters of the µ l -action on O V? The response is most unlucky: Example I.IV.2. Let µ l act on k[x, y] by way of x T x, y T l 1 y. The ring of invariants is R = k[x l, y l, xy] whence the action of C 2 on the plane by way of x y, y x naturally descends to an action on R. Observe however that if l 2 the weighted algebra induced by these characters, (1, l 1), is not invariant under C 2. Indeed the weighted filtration by ideals is I k = (x a y b a + (l 1)b k) k[x, y] and this filtration is clearly not invariant under C 2. From a global perspective, say on a smooth algebraic variety X endowed with a µ l - action, the example is telling us that local choice of characters (étale locally around each fixed point), is by no means sufficient to deduce that the corresponding weighted algebras will patch to a sheaf of algebras on X. With the aim of remedying this lack of uniqueness, it turns out that the existence of a divisor, across which local stabilizers act faithfully, as in definition I.III.4.(ii), is enough to insist that the local weighted algebras glue. Lemma I.IV.3. Let O be a strictly henselian local ring with diagonal cyclic quotient singularities, D an irreducible Q-cartier divisor on Spec(O), and f O V a generator of D V. Assume that the stabilizer acts faithfully across D. Then there exists a unique choice of generator, T, such that µ l acts by way of f T f on Spec(O V ). Proof. The existence of such T is tautologically equivalent to the stabilizer acting faithfully, while the uniqueness is obvious. At this point we can write characters for the µ l -action, profitting from the choice of T as in the previous lemma. In particular, notation as in definition I.III.1, setting x n := f, we obtain characters (a 1,..., a n 1, 1), and correspondingly a weighted algebra B(D) := B(a 1,..., a n 1, 1) over Spec(O). Uniqueness propagates extremely well, and we obtain the fundamental:

12 12 FEDERICO BUONERBA Proposition I.IV.4. Let X be an algebraic variety with diagonal cyclic quotient singularities, D a Q-Cartier divisor which is non-empty, reduced and irreducilble on each connected component of X. Assume that, for every geometric point x D, the local stabilizers act faithfully across D. Then there exists a sheaf of graded algebras B on X, restrcting to B(D x ) étale-locally around each geometric point x. Proof. This is a corollary of the uniqueness statement from lemma I.IV.3. I.V. Toric varieties. Here we recall basic notation of the theory of toric varieties, following [KKMS73, Chap 1]. The scope of this section is to give a combinatorial interpretation of diagonal cyclic quotient singularities, and re-interpret the observations from sections I.III, I.IV in this perspective. Let V be an n-dimensional real vector space. The choice of a basis defines an isomorphsim i : V R n, and consequently a lattice Λ V. Let σ V be a cone which is generated by finitely many elements in Λ, i.e. a toric cone. Such toric cone defines an affine toric variety X σ := Spec(k[σ Λ ]) where k[σ Λ ] is the algebra generated by monomials x a for a σ Λ. Explicitly, a = (a 1,..., a n ) σ Z n σ Λ. It is easy to see that toric open subsets of X σ correspond similarly to the faces of σ; in particular, if τ V is a fan - a polyhedron which is obtained as union of finitely many, finitely generated cones intersecting along common faces - it defines a toric variety X τ, obtained by glueing the affine toric varieties corresponding to the cones defining τ. Crucially, blow-ups can be understood in terms of the toric cone: given a toric cone σ =< v 1,..., v m > and a vector v σ which is not in m i=1 Nv i, we can consider the fan σ v obtained as the union of the cones σ i =< v 1,..., v i 1, v, v i+1,..., v m > Then there exists a sheaf of ideals I v in X σ such that Proj Xσ (I v ) norm Xσv and the toric morphism induced by the inclusion σ v σ corresponds to the natural projection Proj Xσ (I v ) norm X σ. Now we turn our attention to the class of diagonal cyclic quotient singularities by observing that, as in [KKMS73, pp.16-18], those can be described in terms of toric geometry:

13 FUNCTORIAL RESOLUTION OF TAME QUOTIENT SINGULARITIES 13 Proposition I.V.1. Let e 1,..., e n be a basis of a real n-dimensional vector space V and let a 1,..., a n 1, l be natural numbers, l 0. Then the cone σ generated by e 1,..., e n 1, le n (a 1 e a n 1 e n 1 ) defines a toric variety isomorphic to the quotient of A n k by the µ l-action with characters (a 1,..., a n 1, 1). Proof. As above, the basis defines an isomorphism V R n. Under this identification, let N Z n be the subgroup of index l generated by e 1,..., e n 1, le n (a 1 e a n 1 e n 1 ) and let M be its dual lattice in V. Observe that σ is generated by e i + a i l 1 e n (i n 1), l 1 e n which is basis of M, whence Spec(k[σ M]) A n k with coordinates x 1,..., x n corresponding to this choice of generators for σ. Consider now the µ l -action on Spec(k[σ M]) with characters (a 1,.., a n 1, 1). On monomials the action is given by x m T lm(en) x m (m σ M) whence the fixed subalgebra is generated by monomials x m such that m(e n ) Z, which is evidently k[σ Z n ]. We finally conclude our introductory chapter, by proving the existence of local resolution of germs of diagonal cyclic quotient singularities. It is worth pointing out how trivial and explicit this process is, with the notion of weighted / toric blow up at our disposal: Proposition I.V.2. Let µ l act on A n k diagonally with characters (a 1,..., a n 1, 1), and let X be the quotient variety. Let A = A(a 1,..., a n 1, 1) and B = B(a 1,..., a n 1, 1), as in definition I.IV.1. Then Proj X (B) has only diagonal cyclic quotient singularities of orders a 1,..., a n 1. Moreover the modification Proj X (B) X coincides with the toric blow-up obtained by way of the decomposition of σ, as defined in I.V.1, by adding the vector e n.

14 14 FEDERICO BUONERBA Proof. By the local description given at the beginning of I.IV, Proj A n k (A) has diagonal cyclic quotient singularities of orders a 1,..., a n 1. The action of µ l lifts to Proj A n k (A), and it acts by pseudoreflections across the exceptional divisor of Proj A n(a) A n k k. It follows that Proj A n k (A)/µ l = Proj Xσ (B) has diagonal cyclic quotient singularities of orders a 1,..., a n 1. Inspection of the local structure of Proj Xσ (B) around the exceptional divisors reveals immediately that the said toric blow-up gives the same output. II. Resolution The rest of the manuscript will be devoted to the proof of the main theorem, as stated in the introduction. II.I. Step 1: Reduction to tame cyclic quotient singularities. The invariant on which the inductive argument builds upon is i(x) = max x X G x G x being the geometric stabilizer, as in definition I.II.1. By corollary I.I.3 the subvariety Z consisting of points with geometric stabilizer of maximal cardinality is smooth, so let p Z : Y = Bl Z X X be its blow-up with exceptional divisor E. If i(y ) < i(x) we are done, otherwise necessarily i(y ) = i(x) and E is Q-Cartier, whence there is a Cartification cover q E : Y (E) Y. The assumption i(y ) = i(x) implies that q E is not an isomorphism, equivalently E is not Cartier, and in fact more is true: Claim II.I.1. i(y (E)) < i(y ). Proof. Consider the natural projection q V : Y V Y (E) and let y Y V be a geometric point with G y = i(y ). The projection induces a morphism of strictly henselian local rings at y, O Y (E),qV (y) O Y V,y. The action of G y on O Y V,y leaves E y invariant, so by proposition I.I.4 it is given by a non-trivial character χ E,y : G y k. We see immediately that O Y,qE q V (y) is the subring of O Y (E),qV (y) of elements fixed by the cyclic action of Im(χ E,y ). Thus G qv (y) = ker(χ E,y ) and the claim follows. In order to proceed, we need a:

15 FUNCTORIAL RESOLUTION OF TAME QUOTIENT SINGULARITIES 15 Lemma II.I.2. Assume the main theorem holds for quasi-projective varieties X such that i(x) N. Let X be an Artin stack with tame quotient singularities such that i(x ) N. Then there exists a birational and projective morphism X X with X smooth. Proof. We can represent X as a smooth groupoid (s, t) : R X such that i(x) = i(r) N. Functoriality under smooth localization gives unique isomorphisms φ s : M(R) s M(X) and φ t : M(R) t M(X), and the induced morphisms (s φ s, t φ t ) : M(R) M(X) inherit canonically the structure of a smooth groupoid, which is our X. Therefore we can apply our inductive hypothesis, in the formulation for stacks, as in lemma II.I.2, and obtain a birational and projective morphism r Y : Y 1 Y (E) with Y 1 smooth Deligne-Mumford stack, along with a GC quotient morphism m : Y 1 X 1, [KM97, 1.1]. Since Y is quasi-projective and r Y is projective, X 1 is quasi-projective. The total pullback of r Y q E E naturally descends to a Q-Cartier divisor E 1 on X 1, i.e. m E 1 = r Y q E E. The pair (X 1, E 1 ) enjoys particularly good properties, that we can summarize as follows: Claim II.I.3. Y 1 = X 1 (E 1 ). Proof. In our setting the GC quotient morphism m : Y 1 X 1 is locally a cyclic cover, say given by the tame cyclic action of C y in a neighborhood of a geometric point y Y 1. The assertion we need to prove is that C y acts faithfully across r Y q E E. Observe that C y is faithful on q E E Y (E) in a neighborhood of r Y (y), and since Y 1 Y (E) is birational, whence injective on local rings, we conclude by proposition I.I.5 that C y is faithful on ry q EE as well. It is pretty straightforward to control the behavior of our construction under smooth base change: Proposition II.I.4. Let f : X X be a smooth morphism, then the square is fibered. X 1 X 1 X X

16 16 FEDERICO BUONERBA Proof. Consider the diagram smooth Resolution GC quotient X Y = Bl f ZX Y (f E) Y 1 X 1 X Y = Bl Z (X) Y (E) Resolution Y 1 GC quotient X 1 The two leftmost squares are fibered, by lemma I.II.7. The third square from the left is also fibered, by our inductive hypothesis, thus Y 1 = Y 1 X X, and we conclude since the GC quotient functor commutes with flat, hence smooth, base change - as it follows by definition, [KM97, 1.8]. We conclude the first step of the proof by summarizing what we got so far. Summary II.I.5. : Let P k denote the category of pairs (X, E) where X is a quasiprojective variety with tame cyclic quotient singularities, and E is a Q-Cartier divisor such that the canonical projection X V X(E) is an isomorphism. Then there exists a functor N : Q k P k, given on objects by X (X 1, E 1 = E 1 (X)), and a k-morphism p X : X 1 X which is birational and projective. The functor commutes with smooth base change, that is to say for every smooth morphism f : Y X there is a unique Y -isomorphism ψ f : f X 1 Y1 such that ψf p Y E 1(Y ) = p X E 1(X). Here is a diagram summarizing our construction: qv q E E Y V Vistoli qe E Y (E) Cartification resolution m E 1 Y 1 GC quotient=cartification E Bl Z X = Y E 1 X 1 Z X II.II. Step 2: Resolution of diagonal cyclic quotient singularities. The second step of the proof works out an algorithmic procedure to reduce the invariant i(x) all the way down to zero, if we are given a pair N(X) = (X, E), where X is a quasi-projective variety with tame cyclic quotient singularities, E is a reduced divisor whose connected components

17 FUNCTORIAL RESOLUTION OF TAME QUOTIENT SINGULARITIES 17 are irreducible, and the canonical projection X V start, notice: X(E) is an isomorphism. Before we Remark II.II.1. If X V µ l,x acts faithfully across E. X(E), then X sing E, and for every x X, the local stabilizer Which points to the key observation we need to construct a resolution, that for such a pair (X, E) there exists a choice of characters for the cyclic stabilizers, whose corresponding local weighted algebras form a globally well defined sheaf of algebras, proposition I.IV.4. Blowing such sheaf of algebras up, as in proposition I.V.2, we obtain a quasi-projective variety with cyclic quotient singularities whose stabilizers have smaller order. The drawback is, of course, that these need not be tame anymore. On the other hand, cyclic quotients need not be tame for weighted blow-ups to make sense, whence further applications of proposition I.V.2 will eventually lead to a resolution of the singularities. The good way of formalizing the above intuition is the following resolution algorithm for diagonal cyclic quotient singularities, which in turn is remarkably close to that of toroidal singularities, [KKMS73]: Lemma II.II.2. Let D k denote the category of pairs (X, D) where X is a quasi-projective variety with diagonal cyclic quotient singularities and D = t i=1 D i is a reduced divisor. Assume moreover that, if x X sing, then x D and µ l,x is faithful across at least one irreducible component of D. Then there exists a functor M : D k Sm k, and a morphism u X : M(X, D) X which is birational and projective. The functor commutes with smooth base change, that is to say for every smooth morphism f : (Y, D Y ) (X, D) there is a unique Y -isomorphism φ f : f M(X, D) M(Y, D Y ). Proof. Let (X, D) Ob(D k ) and j(x) = max x X µ l,x Reasoning as in corollary I.I.3, with our assumption that X has diagonal cyclic quotient singularities, it is clear that is a smooth subvariety of D. Z = {x X, µ l,x = j(x)} We will proceed by induction on j(x). Let Y be a connected component of Z: denote by h(y ) the maximal index such that Y D h(y ), and such that the generic stabilizer µ l,y is

18 18 FEDERICO BUONERBA faithful across D h(y ). By proposition I.IV.4 for every such Y there is a sheaf of algebras B Y supported on Y, induced by the faithful action of the geometric stabilizers across D h(y ). Let B Z = Y B Y, and denote by p 1 : X 1 = Proj BZ X X with reduced exceptional divisor D t+1, and set D 1 := (p 1 D) red = Di 1 + D t+1, where of course Di 1 is the strict transform of D i under p 1. By proposition I.V.2 j(x 1 ) < j(x), and we claim that the pair (X 1, D 1 ) Ob(D k ). Indeed let Y 1 be a connected component of Z 1. We have two cases: if Y 1 is not contained in D t+1 then Y 1 is the proper transform of some component Y X, and faithfulness across Dh(Y 1 ) is granted by I.I.5. if Y 1 D t+1, then the local description of a weighted blow-up, as given at the beginning of I.IV, forces the geometric stabilizers along Y 1 to be faithful across D t+1. By induction on j(x) we conclude the existence of M(X, D) and u X. Functoriality under smooth localization is a trivial consequence of our iterative construction. II.III. Final assembly. We are ready to conclude the proof of the main theorem: starting with a quasi-projective variety with tame quotient singularities X, section II.I gives a pair N(X) = (X 1, E 1 ) as in summary II.I.5. Running section II.II with input (X 1, E 1 ) - which is possible by remark II.II.1 - we get a smooth quasi-projective variety M(X 1, E 1 ), along with a birational and projective morphism M(X 1, E 1 ) X. Every step commutes with smooth localization, so the same holds for the resolution functor R := M N Let s conclude with an important corollary: Corollary II.1. Let X be an Artin stack with tame quotient singularities. Then there exists a birational and projective morphism X X with X smooth. Proof. See the proof of lemma II.I.2. References [AV01] D. Abramovich, A. Vistoli, Compactifying the space of stable maps, J. Amer. Math. Soc. 15 (2002) pp S / [Be14] D. Bergh, Functorial destackification of tame stacks with abelian stabilisers, org/abs/

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