Applications

To illustrate how the destackification theorem may be applied, we will study three corollaries. The proofs given here will be sketchy, since a more detailed account will appear later in a joint paper with David Rydh.

The destackification algorithm is useful even if one is not primarily interested in stacks. Let $X$ be a variety over a field $k$ whose singular points are all simplicial, toric singularities. By this we mean that each point $\unicode[STIX]{x1D709}\in X$ has an étale neighbourhood $X^{\prime }\rightarrow X$ with $X^{\prime }=U/\unicode[STIX]{x1D6E5}$ for some smooth variety $U$ and finite diagonalisable group $\unicode[STIX]{x1D6E5}$ . In this situation, there exists a canonical stack $X_{\text{can}}$ which is smooth and has $X$ as coarse space [Reference VistoliVis89, Reference SatrianoSat12]. By applying the functorial destackification algorithm on $X_{\text{can}}$ , we obtain a functorial desingularisation algorithm.

Corollary 1.3 (Functorial desingularisation of simplicial toric singularities).

Let $X$ be an algebraic space of finite type over an arbitrary field $k$ . Assume that $X$ has simplicial toric singularities only. Then there exists a sequence

$$\begin{eqnarray}\unicode[STIX]{x1D6F1}:X_{m}\rightarrow \cdots \rightarrow X_{0}=X\end{eqnarray}$$

of proper birational modifications such that $X_{m}$ is smooth. The construction is functorial with respect to change of base field and with respect to smooth morphisms $X^{\prime }\rightarrow X$ .

Note that no toroidal structure is needed. This makes the corollary more general than the toroidal methods described in [Reference Kempf, Knudsen, Mumford and Saint-DonatKKMS73]. On the other hand, the methods described in this article are somewhat less explicit. It recently came to the author’s knowledge that a similar result was obtained in the recent preprint [Reference BuonerbaBuo15].

At first sight, the assumption in Theorem 1.2 that the stack $X$ has diagonalisable stabilisers seems to be quite restrictive. But at least if we work over a field of characteristic 0, this can be overcome. By first using functorial embedded desingularisation on the stacky locus of $X$ with the Bierstone–Milman variant of Hironaka’s method [Reference Bierstone and MilmanBM97], we reduce to the case where the stacky locus is contained in a simple normal crossings divisor. But this implies that the stabilisers are in fact diagonalisable [Reference Reichstein and YoussinRY00, Theorem 4.1], so we are in a situation where we can apply Theorem 1.2.

Finally, destackification can be used to obtain a stacky version of the weak factorisation theorem by Włodarczyk [Reference WłodarczykWło00] for Deligne–Mumford stacks in characteristic 0. The corollary is obtained by applying Włodarczyk’s result on the algebraic space obtained after destackifying using Corollary 1.4.

Outline of the paper

Section 2 collects some preliminaries on algebraic stacks and clarifies the terminology used in this paper. We will also make precise definitions of certain terms, such as functoriality and blow-up sequence, used in the main theorems. In § 3, we will review some basic facts about toric stacks. These will be used in § 4, where we describe two algorithms, Algorithms A and B, which prove the destackification theorems in the toric case. The algorithms are based on the classical toric desingularisation algorithm, but have an additional twist in order to make the process functorial.

From § 5 and onwards, we leave the realm of toric stacks and work with more general smooth stacks with finite diagonalisable stabilisers. First we show that any such stack is locally toric, which allows us to work with local homogeneous coordinates. Then, in § 6, we introduce an invariant, which we call the conormal representation. This invariant captures the local structure of a stack near each point. In characteristic 0, we could have worked with the canonical action of the stabiliser on the tangent space at each point, but, in positive characteristic, the tangent space is not well behaved. Instead, we work with the conormal bundle of the residual gerbe. We will also study a framework for constructing special purpose invariants, called conormal invariants, based on the conormal representation. Simple, well-known examples of such invariants are the order of the stabiliser and the multiplicity of the toric singularity of the corresponding point in the coarse space.

In § 7, we give an outline of the general destackification algorithms and introduce all conormal invariants used by these algorithms. Finally, in § 8, we go through the actual destackification algorithms and prove their correctness.

The paper also includes three appendices, collecting results of more general interest. In Appendix A, we prove a structure theorem for smooth, tame stacks in the spirit of the general structure theorem given in [Reference Abramovich, Olsson and VistoliAOV08]. We will also simplify parts of the proof of the general structure theorem given in [Reference Abramovich, Olsson and VistoliAOV08]. In Appendix B, we compute the cotangent complex of a basic toric stack and, in Appendix C, we give an alternative interpretation of the conormal representation in terms of the cotangent complex.

2.1 Preliminaries and basic terminology

The cyclic (abstract) group of order $n$ will be denoted by $C_{n}$ .

We will use the definitions of algebraic stack and algebraic space used in the Stacks project [Sta17]. By a sheaf, we mean a sheaf on the site of schemes with coverings given by faithfully flat morphisms which are locally of finite presentation, i.e., the fppf topology. By a stack, we mean a stack in groupoids over the same site. An atlas for a stack $X$ is a 1-morphism $f:U\rightarrow X$ , where $U$ and $f$ are representable by algebraic spaces and $f$ is faithfully flat and locally of finite presentation. If the morphism $f$ is smooth, we call it a smooth atlas. A stack is algebraic if it admits an atlas, and it is a theorem that every algebraic stack admits a smooth atlas.

Let $X$ be an algebraic stack. A morphism $\unicode[STIX]{x1D70B}:X\rightarrow X_{\text{cs}}$ is called a coarse space if it is initial among morphisms to algebraic spaces and the induced map $|\unicode[STIX]{x1D70B}|:|X|\rightarrow |X_{\text{cs}}|$ between topological spaces is a homeomorphism. Usually, this is called a coarse moduli space, but we drop the word moduli since we are discussing algebraic stacks without having any specific moduli problem in mind. Due to a classical theorem by Keel and Mori [Reference Keel and MoriKM97] with generalisations by Rydh [Reference RydhRyd13], an algebraic stack $X$ has a coarse space if its inertia stack is finite over  $X$ .

Let $X$ be an algebraic stack which is quasi-separated and locally of finite presentation over a base scheme $S$ . Following Abramovich et al. [Reference Abramovich, Olsson and VistoliAOV08], we say that $X$ is tame if it has finite inertia and linearly reductive stabilisers. This property is reviewed in Appendix A. We will be particularly interested in the case when $X$ has diagonalisable stabilisers. We will use the term orbifold, in the relative sense, for a tame stack $X\rightarrow S$ which is smooth over the base scheme, and which has fibrewise generically trivial stabilisers.

We recall the definition of a simple normal crossings divisor on a regular scheme $X$ . Let $E=E^{1}+\cdots +E^{r}$ be an effective Cartier divisor on $X$ . Given a point $\unicode[STIX]{x1D709}\in X$ , we consider the stalk ${\mathcal{O}}_{X,\unicode[STIX]{x1D709}}$ with maximal ideal $\mathfrak{m}_{\unicode[STIX]{x1D709}}$ . Each divisor $E^{i}$ defines a regular element $f_{\unicode[STIX]{x1D709}}^{i}\in {\mathcal{O}}_{X,\unicode[STIX]{x1D709}}$ which is well defined up to multiplication by a unit. We say that $E$ is a simple normal crossings divisor if it can be written as a sum $E=E^{1}+\cdots +E^{r}$ of Cartier divisors such that at each point $\unicode[STIX]{x1D709}\in X$ , the subsequence of regular parameters $f_{\unicode[STIX]{x1D709}}^{i}$ satisfying $f_{\unicode[STIX]{x1D709}}^{i}\in \mathfrak{m}_{\unicode[STIX]{x1D709}}$ extends to a regular system of parameters $(t_{\unicode[STIX]{x1D709}}^{1},\ldots ,t_{\unicode[STIX]{x1D709}}^{n})$ . A closed subscheme $Z$ of $X$ is said to have simple normal crossings with $E$ if at each point $\unicode[STIX]{x1D709}\in X$ , the system $(t_{\unicode[STIX]{x1D709}}^{1},\ldots ,t_{\unicode[STIX]{x1D709}}^{n})$ described above can be chosen such that the ideal corresponding to $Z$ in ${\mathcal{O}}_{X,\unicode[STIX]{x1D709}}$ is generated by a subsequence of $(t_{\unicode[STIX]{x1D709}}^{1},\ldots ,t_{\unicode[STIX]{x1D709}}^{n})$ .

The definitions above generalise directly to stacks. Let $X$ be a regular algebraic stack with a smooth atlas $\unicode[STIX]{x1D70B}:U\rightarrow X$ . An effective Cartier divisor $E$ is a simple normal crossings divisor if $E$ can be written as a sum $E^{1}+\cdots +E^{r}$ of regular divisors and the pull-back $E_{U}$ to $U$ is a simple normal crossings divisor. Similarly, a closed substack $Z$ has simple normal crossings with $E$ if the same holds for $Z_{U}$ with respect to $E_{U}$ .

We will also consider the relative version of the definitions above. Let $X\rightarrow S$ be a smooth stack over a scheme and let $E=E^{1}+\cdots +E^{r}$ be an effective Cartier divisor on $X$ , with each $E^{i}$ smooth over $S$ . Note that $E$ is a relative effective Cartier divisor in the sense of [Reference Dieudonné and GrothendieckEGAIV, § 21.15]. We say that $E$ is a (relative) simple normal crossings divisor if $E_{\unicode[STIX]{x1D709}}$ is a simple normal crossings divisor on the fibre $X_{\unicode[STIX]{x1D709}}$ for each geometric point $\unicode[STIX]{x1D709}:\text{Spec}\,\bar{k}\rightarrow S$ . Similarly, we define what is meant for a substack $Z\subset X$ which is smooth over $S$ to have simple normal crossings with $E$ .

Note that the term component in this context does not refer to connected component; the components of $\boldsymbol{E}$ , as in the definition above, may well be empty or disconnected.

When referring to the ordering of the components of an ordered simple normal crossings divisor, we will use an age metaphor. The components of such a divisor form a sequence $E^{1}\!,\ldots ,E^{r}\!$ . The indices may be thought of as birth dates of the components, and we say that $E^{i}$ is older than $E^{j}$ , and that $E^{j}$ is younger than $E^{i}$ , provided that $i<j$ .

2.2 Stacky blow-up sequences

Let $S$ be a scheme and let $(X,\boldsymbol{E})/S$ be a standard pair. By a smooth blow-up of $(X,\boldsymbol{E})/S$ , we mean a blow-up $\unicode[STIX]{x1D70B}:\text{Bl}_{Z}X\rightarrow X$ in a centre $Z$ which is smooth over $S$ and has simple normal crossings with $E$ . The transform of $(X,\boldsymbol{E})/S$ along $\unicode[STIX]{x1D70B}$ is the pair $(\text{Bl}_{Z}X,\boldsymbol{E}^{\prime })$ , where $\boldsymbol{E}^{\prime }$ denotes the ordered set of the strict transforms of the components of $\boldsymbol{E}$ followed by the exceptional divisor. In other words, the ordering of the components $\boldsymbol{E}^{\prime }$ is the one induced by the ordering on $\boldsymbol{E}$ with the exceptional divisor added as the youngest divisor.

The root construction of a stack in an effective Cartier divisor is thoroughly described in for instance [Reference Abramovich, Graber and VistoliAGV08, Reference CadmanCad07] and [Reference Fantechi, Mann and NironiFMN10, § 1.3.b]. Let $(X,\boldsymbol{E})/S$ be a standard pair. We will only consider root stacks with roots taken of components of $\boldsymbol{E}$ . Such a root stack will be called a smooth root stack. If $E^{i}\in \boldsymbol{E}$ , we use the notation $X_{d^{-1}E^{i}}\rightarrow X$ for the $d$ th root of $E^{i}$ . If $\boldsymbol{d}$ is a sequence of positive integers indexed by the elements of $\boldsymbol{E}$ , then $X_{\boldsymbol{d}^{-1}\boldsymbol{E}}\rightarrow X$ denotes the fibre product of the stacks $X_{d_{i}^{-1}E^{i}}$ over $X$ for all $E^{i}\in \boldsymbol{E}$ with $d_{i}$ as corresponding element in $\boldsymbol{d}$ . The sum of the components of $\boldsymbol{E}$ for which the corresponding index in $\boldsymbol{d}$ is larger than one is called the centre of the root stack. If $\boldsymbol{E}^{\prime }$ is a subsequence of $\boldsymbol{E}$ and $\boldsymbol{d}$ is a sequence of positive integers indexed by $\boldsymbol{E}^{\prime }$ , then we sometimes use the notation $X_{\boldsymbol{d}^{-1}\boldsymbol{E}^{\prime }}\rightarrow X$ for the root stack obtained by extending $\boldsymbol{d}$ to $\boldsymbol{E}$ by ones. The transform of a pair $(X,\boldsymbol{E})/S$ along a smooth root stack $\unicode[STIX]{x1D70B}:X_{\boldsymbol{d}^{-1}\boldsymbol{E}}\rightarrow X$ is the pair $(X_{\boldsymbol{d}^{-1}\boldsymbol{E}},\unicode[STIX]{x1D70B}^{-1}\boldsymbol{E})$ . Here $\unicode[STIX]{x1D70B}^{-1}\boldsymbol{E}$ denotes the sequence $(d_{i}^{-1}\unicode[STIX]{x1D70B}^{\ast }E^{i})$ of roots of the components $E^{i}$ of  $\boldsymbol{E}$ .

A smooth stacky blow-up is either a smooth root stack or a smooth blow-up. Note that the transform of a standard pair along a smooth stacky blow-up is again a standard pair.

Definition 2.2. Let $(X_{0},\boldsymbol{E}_{0})/S$ be a standard pair. A smooth, stacky blow-up sequence of $(X_{0},\boldsymbol{E}_{0})/S$ of length $n$ is a sequence

$$\begin{eqnarray}\unicode[STIX]{x1D6F1}:(X_{n},\boldsymbol{E}_{n})\stackrel{\unicode[STIX]{x1D70B}_{n}}{\longrightarrow }\cdots \stackrel{\unicode[STIX]{x1D70B}_{1}}{\longrightarrow }(X_{0},\boldsymbol{E}_{0}),\end{eqnarray}$$

where each $\unicode[STIX]{x1D70B}_{i}$ , for $1\leqslant i\leqslant n$ , is a smooth stacky blow-up in a centre $Z_{i-1}$ and each $(X_{i},\boldsymbol{E}_{i})$ is the transform of $(X_{i-1},\boldsymbol{E}_{i-1})$ along $\unicode[STIX]{x1D70B}_{i}$ . The centres $Z_{i}$ for $0\leqslant i\leqslant n-1$ , although suppressed from the notation, are considered part of the structure. We require each $Z_{i}$ to have positive codimension in $X_{i}$ at each of its points. If all stacky blow-ups are in fact usual blow-ups, we call $\unicode[STIX]{x1D6F1}$ a smooth, ordinary blow-up sequence.

Since all blow-up sequences we consider in this article will be smooth, stacky blow-up sequences, we will usually drop the modifiers smooth and stacky and just say blow-up sequence.

Definition 2.3. Let $(X_{0},\boldsymbol{E}_{0})/S$ be a standard pair and let

$$\begin{eqnarray}\unicode[STIX]{x1D6F1}:(Y,\boldsymbol{F})=(X_{n},\boldsymbol{E}_{n})\xrightarrow[{}]{\unicode[STIX]{x1D70B}_{n}}\cdots \xrightarrow[{}]{\unicode[STIX]{x1D70B}_{1}}(X_{0},\boldsymbol{E}_{0})\end{eqnarray}$$

be a smooth, stacky blow-up sequence on $(X_{0},\boldsymbol{E}_{0})/S$ . Let $\unicode[STIX]{x1D70B}:Y\rightarrow Y_{\text{cs}}$ be the coarse space. We call $\unicode[STIX]{x1D6F1}$ a destackification if the following conditions hold.

1. (i) The space $Y_{\text{cs}}$ is smooth over $S$ .

2. (ii) Each component of $\boldsymbol{F}_{\text{cs}}=(F_{\text{cs}}^{i}\mid F^{i}\in \boldsymbol{F})$ is smooth and $\sum _{i}F^{i}$ has simple normal crossings.

3. (iii) The divisor $\boldsymbol{F}$ is a $\boldsymbol{d}$ th root of the pull-back $\unicode[STIX]{x1D70B}^{\ast }\boldsymbol{F}_{\text{cs}}$ for some sequence $\boldsymbol{d}$ of positive integers indexed by the components of $\boldsymbol{F}$ .

4. (iv) The canonical factorisation $Y\rightarrow (Y_{\text{cs}})_{\boldsymbol{d}^{-1}\boldsymbol{F}_{\text{cs}}}\rightarrow Y_{\text{cs}}$ through the root stack makes $Y$ a gerbe over $(Y_{\text{cs}})_{\boldsymbol{d}^{-1}\boldsymbol{F}_{\text{cs}}}$ . In particular, if $X_{0}$ is an orbifold, then $Y\rightarrow (Y_{\text{cs}})_{\boldsymbol{d}^{-1}\boldsymbol{F}_{\text{cs}}}$ is an isomorphism.

The conditions (i) and (ii) can be summarised by saying that the pair $(Y_{\text{cs}},\boldsymbol{F}_{\text{cs}})/S$ is a standard pair.

A stacky blow-up is said to be empty if the centre is empty. Although the algorithms used in the constructions mentioned in the main theorems will never produce blow-up sequences containing empty blow-ups, such may occur after pulling back blow-up sequences along morphisms which are not surjective. We will consider such pull-backs when discussing functoriality below. We regard two blow-up sequences $\unicode[STIX]{x1D6F1}$ and $\unicode[STIX]{x1D6F1}^{\prime }$ to be equivalent if, after pruning them from empty blow-ups, they fit into a 2-commutative ladder

such that the vertical morphisms are isomorphisms preserving the centres.

2.3 Gerbes

Given a stack $X$ , we denote by $X\rightarrow X_{\text{sh}}$ the canonical morphism to the coarse sheaf. This is, by definition, a morphism which is initial among morphisms to sheaves in the fppf topology. An algebraic stack $X$ is called a gerbe if its associated coarse sheaf is an algebraic space. In this case, the morphism $\unicode[STIX]{x1D70B}:X\rightarrow X_{\text{sh}}$ is called the structure morphism of the gerbe. There is also a relative notion of gerbe. A morphism $\unicode[STIX]{x1D70B}:X\rightarrow Y$ of algebraic stacks is called a gerbe if the base change $X^{\prime }\rightarrow Y^{\prime }$ of $\unicode[STIX]{x1D70B}$ along any morphism $Y^{\prime }\rightarrow Y$ , with $Y^{\prime }$ an algebraic space, is the structure morphism of a gerbe in the sense described above.

This definition is standard and coincides with the one given in, for instance, the Stacks project [Sta17]. It is related to the definition given by Giraud [Reference GiraudGir71, Definition 2.1.1] in the following way. A stack $Y$ has a canonical structure of a site induced by the fppf topology on the site of schemes. A morphism $\unicode[STIX]{x1D70B}:X\rightarrow Y$ of algebraic stacks is a gerbe in our sense if and only if $X$ is a gerbe in the sense of Giraud when viewed as a stack in groupoids over the site $Y$ .

Note that in the context of algebraic stacks, the terminology for gerbes sometimes conflicts with algebrogeometric notions. For instance, the structure morphism $\unicode[STIX]{x1D70B}:X\rightarrow Y$ of a gerbe in our sense is always smooth (see Proposition A.2). In particular, the morphism $\unicode[STIX]{x1D70B}$ always admits a section étale locally on $Y$ . This does, however, not imply that $X$ is a gerbe as a stack in groupoids over $Y$ considered as a site endowed with the smooth or étale topology. This is a potential source of confusion as it makes terms such as smooth gerbe ambiguous. In this article, we try to avoid the confusion by consistently working with the fppf topology.

2.4 Stabiliser-preserving maps

We recall the definition and some basic facts about stabiliser-preserving 1-morphisms of stacks. Let $f:X\rightarrow Y$ be a 1-morphism of stacks. Given a generalised point $\unicode[STIX]{x1D709}:T\rightarrow X$ , where $T$ is a scheme, we get an induced map of stabilisers $\text{Stab}_{\unicode[STIX]{x1D709}}X\rightarrow \text{Stab}_{f\circ \unicode[STIX]{x1D709}}Y$ over $T$ . The map $f$ also induces a pair of 2-commutative diagrams

Here $I_{X}\rightarrow X$ denotes the inertia stack of $X$ , and the map $X\rightarrow X_{\text{sh}}$ is the coarse sheaf of $X$ (as defined in the previous section).

In particular, monomorphisms between stacks are stabiliser-preserving. Note that the notions of stabiliser-preserving and point-wise stabiliser-preserving maps are distinct. Both properties are preserved under composition and arbitrary base change. They are also local on the base in the fppf topology.

A useful fact is that if $f:X\rightarrow Y$ is an étale map between algebraic stacks with finite inertia, then the locus where $f$ is point-wise stabiliser-preserving is open in $X$ , and $f$ is stabiliser-preserving over this locus [Reference RydhRyd11, Proposition 6.5]. In fact, if the stacks are tame, the corresponding fact for smooth morphisms is also true, but we will not use this here.

2.5 Functoriality

We consider two basic situations when a smooth stacky blow-up sequence can be transferred from one standard pair to another. Fix a standard pair $(X,\boldsymbol{E})/S$ .

The first situation is when we change base scheme. Given a morphism $S^{\prime }\rightarrow S$ , we can form the pull-backs $X_{S^{\prime }}=X\times _{S}S^{\prime }$ and $\boldsymbol{E}_{S^{\prime }}=\boldsymbol{E}\times _{S}S^{\prime }$ . Then the pair $(X_{S^{\prime }},\boldsymbol{E}_{S^{\prime }})/S^{\prime }$ is also standard, and any smooth stacky blow-up sequence on $(X,\boldsymbol{E})/S$ pulls back to a smooth stacky blow-up sequence on $(X_{S^{\prime }},\boldsymbol{E}_{S^{\prime }})/S^{\prime }$ .

The second situation is when we have a morphism of stacks $X^{\prime }\rightarrow X$ which is smooth. Then we can form the pull-back $\boldsymbol{E}^{\prime }=\boldsymbol{E}\times _{X}X^{\prime }$ , and we get a standard pair $(X^{\prime },\boldsymbol{E}^{\prime })/S$ . Again, any smooth stacky blow-up sequence on $(X,\boldsymbol{E})/S$ pulls back to a smooth stacky blow-up sequence on $(X^{\prime },\boldsymbol{E}^{\prime })/S$ .

The constructions in the main theorems are functorial with respect to arbitrary base change $S^{\prime }\rightarrow S$ and with respect to morphisms $X^{\prime }\rightarrow X$ which are either gerbes or smooth and stabiliser-preserving. It is, however, not reasonable to expect the construction to be functorial with respect to any smooth morphism. Indeed, if $X^{\prime }\rightarrow X$ is a smooth atlas, any reasonable destackification of $X^{\prime }$ must be trivial, whereas a destackification of $X$ cannot be trivial in general.

2.6 Distinguished structure

We do not want our algorithms to modify the locus lying over the smooth locus of the coarse space of the original stack. This poses a problem when it comes to root stacks, since they always modify the entire divisor of which the root is taken. Thus, we would like to keep track of divisors which we are allowed to root. We do this by marking certain divisors as distinguished.

The transform $(X^{\prime },\boldsymbol{E}^{\prime })/S$ of an admissible stacky blow-up of a standard pair with distinguished structure $(X,\boldsymbol{E},\boldsymbol{D})/S$ again has a distinguished structure $\boldsymbol{D}^{\prime }\subset \boldsymbol{E}^{\prime }$ . It consists of the strict transforms of all distinguished divisors along with the exceptional divisor.

3.1 Basic toric stacks

First we introduce basic toric stacks. They play a similar role in the theory of toric stacks as affine toric varieties in the theory of toric varieties. Toric stacks in general are obtained by gluing basic toric stacks together along toric morphisms in the Zariski topology. Note that the term basic toric stack is non-standard.

Fix a scheme $S$ . Consider a finitely generated abelian group $A$ and an $A$ -graded quasi-coherent sheaf of ${\mathcal{O}}_{S}$ -algebras ${\mathcal{R}}$ . The $A$ -grading on ${\mathcal{R}}$ corresponds to an action of the Cartier dual $A^{\vee }$ (over $S$ ) on the relative spectrum $\text{Spec}_{{\mathcal{O}}_{S}}{\mathcal{R}}$ . We get an algebraic stack associated to the pair $({\mathcal{R}},A)$ by taking the stack quotient $[\text{Spec}_{{\mathcal{O}}_{S}}{\mathcal{R}}/A^{\vee }]$ .

Let $({\mathcal{R}}^{\prime },A^{\prime })$ be another pair as above. A graded homomorphism $(f,\unicode[STIX]{x1D711}):({\mathcal{R}},A)\rightarrow ({\mathcal{R}}^{\prime },A^{\prime })$ is a pair where $\unicode[STIX]{x1D711}:A\rightarrow A^{\prime }$ is a group homomorphism and $f:{\mathcal{R}}\rightarrow {\mathcal{R}}^{\prime }$ a morphism of sheaves of ${\mathcal{O}}_{S}$ -algebras. The morphism $f$ is required to take homogeneous sections of degree $a\in A$ to homogeneous sections of degree $\unicode[STIX]{x1D711}(a)$ . Such a graded homomorphism induces a morphism $[\text{Spec}_{{\mathcal{O}}_{S}}{\mathcal{R}}^{\prime }/{A^{\prime }}^{\vee }]\rightarrow [\text{Spec}_{{\mathcal{O}}_{S}}{\mathcal{R}}/A^{\vee }]$ of quotient stacks.

It should be noted that the homogeneous coordinate ring does not determine the isomorphism class of the basic toric stack uniquely.

Proposition 3.2. Let $X$ be a basic toric stack with homogeneous coordinate ring $({\mathcal{R}},A,\boldsymbol{a})$ with

$$\begin{eqnarray}{\mathcal{R}}={\mathcal{O}}_{S}[x_{1},\ldots ,x_{r}][x_{s+1}^{-1},\ldots ,x_{r}^{-1}].\end{eqnarray}$$

Let $K$ be the subgroup $\langle a_{s+1},\ldots ,a_{r}\rangle$ and define $r^{\prime }$ as $r$ minus the rank of $K$ . Consider the $A/K$ -graded sheaf of ${\mathcal{O}}_{S}$ -algebras

$$\begin{eqnarray}{\mathcal{R}}^{\prime }={\mathcal{O}}_{S}[x_{1},\ldots ,x_{r^{\prime }}][x_{s+1}^{-1},\ldots ,x_{r^{\prime }}^{-1}]\end{eqnarray}$$

with the grading given by the weight vector $\overline{\boldsymbol{a}}=(\overline{a}_{1},\ldots ,\overline{a}_{r^{\prime }})$ , where $\overline{a}_{i}$ denotes the image of $a_{i}$ in $A/K$ . Then the basic toric stack associated to the triple $({\mathcal{R}}^{\prime },A/K,\overline{\boldsymbol{a}})$ is isomorphic to $X$ .

Proof. The proposition is trivial if all of the weights $a_{s+1},\ldots ,a_{r}$ are zero. Hence, we assume that one of these weights, which we without loss of generality may assume is $a_{r}$ , is non-zero. By induction, it is enough to prove that the stack $X$ is isomorphic to the stack associated to $({\mathcal{R}}^{\prime },A/K,\overline{\boldsymbol{a}})$ with $K=\langle a_{r}\rangle$ and ${\mathcal{R}}^{\prime }$ , $r^{\prime }$ and  $\overline{\boldsymbol{a}}$ being as in the statement of the proposition.

Assume that $a_{r}$ has infinite order. Then it is straightforward to verify that the graded homomorphism ${\mathcal{R}}\rightarrow {\mathcal{R}}^{\prime }$ defined by

$$\begin{eqnarray}x_{i}\mapsto \left\{\begin{array}{@{}ll@{}}1 & \text{if }i=r,\\ x_{i} & \text{otherwise}\end{array}\right.\end{eqnarray}$$

induces an isomorphism of stacks.

Now assume that $a_{r}$ has finite order $m$ . By a simple group theoretical argument, we may find an element $a_{r}^{\prime }\in A$ such that $a_{r}=na_{r}^{\prime }$ for some positive integer $n$ and $A$ splits as $A_{0}\oplus \langle a_{r}^{\prime }\rangle$ . Consider the surjective homomorphism $A^{\prime }=A_{0}\oplus \mathbb{Z}\rightarrow A$ taking $(a,1)$ to $a+a_{r}^{\prime }$ . We endow the sheaf of rings ${\mathcal{R}}[t,t^{-1}]$ with an $A$ -grading, with $|x_{r}|=(1,n)$ , $|t|=(1,nm)$ and $|x_{i}|$ are arbitrary lifts of $a_{i}$ to $A^{\prime }$ for $i\in \{1,\ldots ,r-1\}$ . Then the graded ring homomorphism $({\mathcal{R}}[t,t^{-1}],A^{\prime })\rightarrow ({\mathcal{R}},A)$ taking $x_{i}$ to $x_{i}$ for $i=1,\ldots ,r$ and $t$ to $1$ induces an isomorphism of stacks, since the order of $|t|$ is infinite. But the graded homomorphism $({\mathcal{R}}[t,t^{-1}],A^{\prime })\rightarrow ({\mathcal{R}}^{\prime },A/K)$ taking $x_{i}$ to $x_{i}$ for $i=1,\ldots ,r-1$ , $x_{r}$ to 1 and $t$ to $x_{r}$ also induces an isomorphism, since also $|x_{r}|$ has infinite order in $({\mathcal{R}}[t,t^{-1}],A^{\prime })$ . This concludes the proof.◻

3.2 Toric orbifolds

As with toric varieties, the gluing together of basic toric stacks can be described combinatorially. We review the parts of the theory we need in this article, restricting the discussion to toric orbifolds with no torus factors.

Let $N$ be a lattice of rank $n$ , and consider it as a subset of the vector space $N_{\mathbb{R}}:=N\otimes _{\mathbb{Z}}\mathbb{R}$ . By a cone $\unicode[STIX]{x1D70E}$ in $N_{\mathbb{R}}$ , we will always mean a polyhedral, rational and strictly convex cone. We write $\unicode[STIX]{x1D70F}\preccurlyeq \unicode[STIX]{x1D70E}$ if $\unicode[STIX]{x1D70F}$ is a face of $\unicode[STIX]{x1D70E}$ . By $\unicode[STIX]{x1D70E}(1)$ , we mean the set of one-dimensional faces, also called the extremal rays, of $\unicode[STIX]{x1D70E}$ . Recall that $\unicode[STIX]{x1D70E}$ is called simplicial if the cardinality of $\unicode[STIX]{x1D70E}(1)$ equals the dimension of the subspace of $N_{\mathbb{R}}$ spanned by $\unicode[STIX]{x1D70E}(1)$ .

Given a fan $\unicode[STIX]{x1D6F4}$ in $N_{\mathbb{R}}$ , we denote the set of rays, that is, the set of one-dimensional cones, in $\unicode[STIX]{x1D6F4}$ by $\unicode[STIX]{x1D6F4}(1)$ . A fan is simplicial if all its cones are. We will frequently consider the free abelian group $\mathbb{Z}^{\unicode[STIX]{x1D6F4}(1)}$ on the set of rays in a fan $\unicode[STIX]{x1D6F4}$ . An element $c_{1}\unicode[STIX]{x1D70C}_{1}+\cdots +c_{r}\unicode[STIX]{x1D70C}_{r}\in \mathbb{Z}^{\unicode[STIX]{x1D6F4}(1)}$ , with $c_{i}\in \mathbb{Z}$ and $\unicode[STIX]{x1D70C}_{i}\in \unicode[STIX]{x1D6F4}(1)$ , is called effective if all coefficients $c_{i}$ are greater than or equal to zero.

Given an orbifold fan $\boldsymbol{\unicode[STIX]{x1D6F4}}=(N,\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D6FD})$ , we construct a toric orbifold via the Cox construction. Denote the dual $\text{Hom}_{\mathbb{Z}}(N,\mathbb{Z})$ by $M$ . Then the Cartier dual of $M$ over $S$ is an $n$ -dimensional torus, which we denote by $T_{N}$ . Its cocharacter and character groups may be canonically identified with $N$ and $M$ , respectively. The morphism $\unicode[STIX]{x1D6FD}$ induces a homomorphism of algebraic groups $T_{\mathbb{Z}^{\unicode[STIX]{x1D6F4}(1)}}\rightarrow T_{N}$ , which fits into an exact sequence

$$\begin{eqnarray}1\rightarrow \unicode[STIX]{x1D6E5}(\boldsymbol{\unicode[STIX]{x1D6F4}})\rightarrow T_{\mathbb{Z}^{\unicode[STIX]{x1D6F4}(1)}}\rightarrow T_{N}\rightarrow 1,\end{eqnarray}$$

where the exactness at the term $T_{N}$ is ensured by the fact that $|\unicode[STIX]{x1D6F4}|$ spans $N_{\mathbb{R}}$ . Now consider the lattice $\mathbb{Z}^{\unicode[STIX]{x1D6F4}(1)}$ and the corresponding space $\mathbb{R}^{\unicode[STIX]{x1D6F4}(1)}$ . Given a cone $\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6F4}$ , we have a corresponding cone $\widetilde{\unicode[STIX]{x1D70E}}$ in $\mathbb{R}^{\unicode[STIX]{x1D6F4}(1)}$ spanned by the rays $\unicode[STIX]{x1D70C}\in \unicode[STIX]{x1D70E}(1)$ viewed as generators in $\mathbb{R}^{\unicode[STIX]{x1D6F4}(1)}$ . Collectively, the cones $\widetilde{\unicode[STIX]{x1D70E}}$ , for $\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6F4}$ , form a fan $\widetilde{\unicode[STIX]{x1D6F4}}$ in $\mathbb{R}^{\unicode[STIX]{x1D6F4}(1)}$ . Denote the corresponding toric variety, or rather family of toric varieties over $S$ , by $X_{\widetilde{\unicode[STIX]{x1D6F4}}}$ .

We give an explicit description of the family $X_{\widetilde{\unicode[STIX]{x1D6F4}}}$ of varieties. The total coordinate ring associated to $\unicode[STIX]{x1D6F4}$ is the sheaf of rings ${\mathcal{R}}={\mathcal{O}}_{S}[x_{\unicode[STIX]{x1D70C}}\mid \unicode[STIX]{x1D70C}\in \unicode[STIX]{x1D6F4}(1)]$ . The irrelevant ideal is the ideal sheaf

$$\begin{eqnarray}B(\unicode[STIX]{x1D6F4})=\langle x^{\hat{\unicode[STIX]{x1D70E}}}\mid \unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6F4}\rangle ,\end{eqnarray}$$

where $x^{\hat{\unicode[STIX]{x1D70E}}}$ denotes the product of all elements $x_{\unicode[STIX]{x1D70C}}$ with $\unicode[STIX]{x1D70C}\not \in \unicode[STIX]{x1D70E}(1)$ . Let $\mathbb{A}_{S}^{\unicode[STIX]{x1D6F4}(1)}=\text{Spec}_{{\mathcal{O}}_{S}}{\mathcal{R}}$ be the relative spectrum and $Z(\unicode[STIX]{x1D6F4})$ be the closed subscheme associated to the irrelevant ideal $B(\unicode[STIX]{x1D6F4})$ . The scheme $X_{\widetilde{\unicode[STIX]{x1D6F4}}}$ is simply $\mathbb{A}_{S}^{\unicode[STIX]{x1D6F4}(1)}\setminus Z(\unicode[STIX]{x1D6F4})$ . Note that the torus $T_{\mathbb{Z}^{\unicode[STIX]{x1D6F4}(1)}}$ is embedded in $X_{\widetilde{\unicode[STIX]{x1D6F4}}}$ in a natural way, and the action of $\unicode[STIX]{x1D6E5}(\boldsymbol{\unicode[STIX]{x1D6F4}})$ on $T_{\mathbb{Z}^{\unicode[STIX]{x1D6F4}(1)}}$ extends to $X_{\widetilde{\unicode[STIX]{x1D6F4}}}$ .

Just like in the case with usual toric varieties, there is an order-reversing correspondence between cones in $\boldsymbol{\unicode[STIX]{x1D6F4}}=(N,\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D6FD})$ and orbit closures in $X_{\boldsymbol{\unicode[STIX]{x1D6F4}}}$ . Given a cone $\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6F4}$ , we have a closed variety $V(\langle x_{\unicode[STIX]{x1D70C}},\unicode[STIX]{x1D70C}\in \unicode[STIX]{x1D70E}(1)\rangle )$ in $X_{\widetilde{\unicode[STIX]{x1D6F4}}}$ . Since this closed variety is $\unicode[STIX]{x1D6E5}(\boldsymbol{\unicode[STIX]{x1D6F4}})$ -invariant, it descends to a closed substack $V(\unicode[STIX]{x1D70E})$ of $X_{\boldsymbol{\unicode[STIX]{x1D6F4}}}$ . In the particular case when we have a ray $\unicode[STIX]{x1D70C}\in \unicode[STIX]{x1D6F4}(1)$ , the substack $V(\unicode[STIX]{x1D70C})\subset X_{\boldsymbol{\unicode[STIX]{x1D6F4}}}$ is a prime divisor, and we denote it by $D_{\unicode[STIX]{x1D70C}}$ . The divisor $D_{\unicode[STIX]{x1D70C}}$ is a smooth Cartier divisor. More generally, if $\unicode[STIX]{x1D713}=c_{1}\unicode[STIX]{x1D70C}_{1}+\cdots +c_{r}\unicode[STIX]{x1D70C}_{r}$ is an element of $\mathbb{Z}^{\unicode[STIX]{x1D6F4}(1)}$ , we let $D_{\unicode[STIX]{x1D713}}$ denote the divisor $c_{1}D_{\unicode[STIX]{x1D70C}_{1}}+\cdots +c_{r}D_{\unicode[STIX]{x1D70C}_{r}}$ . Such a divisor is called a toric divisor, and is a simple normal crossings divisor.

3.3 Morphisms of toric orbifolds

Next we describe morphisms of orbifold fans and toric orbifolds. Our definition is different from, but equivalent to, the one given by Iwanari in [Reference IwanariIwa09a].

Recall that a morphism of fans $f:(N,\unicode[STIX]{x1D6F4})\rightarrow (N^{\prime },\unicode[STIX]{x1D6F4}^{\prime })$ is a group homomorphism $f:N\rightarrow N^{\prime }$ such that the induced map $f_{\mathbb{R}}=f\otimes _{\mathbb{Z}}\mathbb{R}$ maps each cone $\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6F4}$ into a cone $\unicode[STIX]{x1D70E}^{\prime }\in \unicode[STIX]{x1D6F4}^{\prime }$ . This extends to orbifold fans as follows.

Definition 3.10. Consider the orbifold fans $\boldsymbol{\unicode[STIX]{x1D6F4}}=(N,\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D6FD})$ and $\boldsymbol{\unicode[STIX]{x1D6F4}}^{\prime }=(N^{\prime },\unicode[STIX]{x1D6F4}^{\prime },\unicode[STIX]{x1D6FD}^{\prime })$ . A morphism $\boldsymbol{\unicode[STIX]{x1D6F4}}\rightarrow \boldsymbol{\unicode[STIX]{x1D6F4}}^{\prime }$ of orbifold fans is a pair $(f,\hat{f})$ of group homomorphisms fitting into a commutative square

such that both $f:(N,\unicode[STIX]{x1D6F4})\rightarrow (N^{\prime },\unicode[STIX]{x1D6F4}^{\prime })$ and $\hat{f}:(\mathbb{Z}^{\unicode[STIX]{x1D6F4}(1)},\widetilde{\unicode[STIX]{x1D6F4}})\rightarrow (\mathbb{Z}^{\unicode[STIX]{x1D6F4}^{\prime }(1)},\widetilde{\unicode[STIX]{x1D6F4}}^{\prime })$ are morphisms of fans. Since $\hat{f}$ is uniquely determined by $f$ , we often omit $\hat{f}$ from the notation, and simply say that $f:\boldsymbol{\unicode[STIX]{x1D6F4}}\rightarrow \boldsymbol{\unicode[STIX]{x1D6F4}}^{\prime }$ is a morphism of orbifold fans.

It is easy to see that a morphism $f:\boldsymbol{\unicode[STIX]{x1D6F4}}\rightarrow \boldsymbol{\unicode[STIX]{x1D6F4}}^{\prime }$ of orbifold fans induces a corresponding equivariant morphism of pairs $(X_{\widetilde{\unicode[STIX]{x1D6F4}}},\unicode[STIX]{x1D6E5}(\boldsymbol{\unicode[STIX]{x1D6F4}}))\rightarrow (X_{\widetilde{\unicode[STIX]{x1D6F4}}^{\prime }},\unicode[STIX]{x1D6E5}(\boldsymbol{\unicode[STIX]{x1D6F4}}^{\prime }))$ , which, in turn, induces a 1-morphism $X_{\boldsymbol{\unicode[STIX]{x1D6F4}}}\rightarrow X_{\boldsymbol{\unicode[STIX]{x1D6F4}}^{\prime }}$ of toric orbifolds. This gives a functor from the category of orbifold fans to the category of orbifolds over a base scheme $S$ , and we call its essential image the category of toric orbifolds (without torus factors).

The simplest example of a toric morphism is that of toric open immersions, which correspond to subfans of stacky fans. Let $\boldsymbol{\unicode[STIX]{x1D6F4}}=(N,\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D6FD})$ be an orbifold fan. A subfan $\boldsymbol{\unicode[STIX]{x1D6F4}}^{\prime }\subset \boldsymbol{\unicode[STIX]{x1D6F4}}$ is a triple $\boldsymbol{\unicode[STIX]{x1D6F4}}^{\prime }=(N,\unicode[STIX]{x1D6F4}^{\prime },\unicode[STIX]{x1D6FD}^{\prime })$ , where $\unicode[STIX]{x1D6F4}^{\prime }\subset \unicode[STIX]{x1D6F4}$ is a subset, which is a fan in its own right, and $\unicode[STIX]{x1D6FD}^{\prime }$ is the restriction of $\unicode[STIX]{x1D6FD}$ to $\mathbb{Z}^{\unicode[STIX]{x1D6F4}^{\prime }(1)}$ . The canonical map $\boldsymbol{\unicode[STIX]{x1D6F4}}^{\prime }\rightarrow \boldsymbol{\unicode[STIX]{x1D6F4}}$ , which is the identity on $N$ , corresponds to an open immersion $X_{\boldsymbol{\unicode[STIX]{x1D6F4}}^{\prime }}\rightarrow X_{\boldsymbol{\unicode[STIX]{x1D6F4}}}$ . We say that $X_{\boldsymbol{\unicode[STIX]{x1D6F4}}^{\prime }}$ is a toric open substack of $X_{\boldsymbol{\unicode[STIX]{x1D6F4}}}$ . The toric substacks corresponding to orbifold fans with a single maximal cone $\unicode[STIX]{x1D70E}\in \boldsymbol{\unicode[STIX]{x1D6F4}}$ are of particular importance. We denote the corresponding substack, which is a basic toric orbifold, by $U_{\unicode[STIX]{x1D70E}}$ .

3.4 Toric stacky blow-ups

For smooth toric varieties, blow-ups at orbit closures correspond to star subdivisions. This generalises to toric orbifolds (see [Reference Edidin and MoreEM12]). We recall the definition here.

If $X_{\boldsymbol{\unicode[STIX]{x1D6F4}}}$ is the toric orbifold corresponding to the orbifold fan $\boldsymbol{\unicode[STIX]{x1D6F4}}$ , and $\unicode[STIX]{x1D70E}$ is a cone in $\boldsymbol{\unicode[STIX]{x1D6F4}}$ , then the map $X_{\boldsymbol{\unicode[STIX]{x1D6F4}}^{\ast }(\unicode[STIX]{x1D70E})}\rightarrow X_{\boldsymbol{\unicode[STIX]{x1D6F4}}}$ corresponding to the star subdivision is the blow-up of $X_{\boldsymbol{\unicode[STIX]{x1D6F4}}}$ with centre $V(\unicode[STIX]{x1D70E})$ . The divisor $D_{\unicode[STIX]{x1D70C}_{0}}$ on $X_{\boldsymbol{\unicode[STIX]{x1D6F4}}^{\ast }(\unicode[STIX]{x1D70E})}$ corresponding to the exceptional ray $\unicode[STIX]{x1D70C}_{0}$ is the exceptional divisor of the blow-up.

Definition 3.14. Let $\boldsymbol{\unicode[STIX]{x1D6F4}}=(N,\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D6FD})$ be an orbifold fan and $\boldsymbol{\unicode[STIX]{x1D70C}}=\{\unicode[STIX]{x1D70C}_{1},\ldots ,\unicode[STIX]{x1D70C}_{r}\}\subset \unicode[STIX]{x1D6F4}(1)$ a set of rays. For each $\unicode[STIX]{x1D70C}_{i}\in \boldsymbol{\unicode[STIX]{x1D70C}}$ , we assign a weight $d_{i}$ , which is a positive integer. Denote the function taking each ray to its weight by $\boldsymbol{d}$ . Consider the group homomorphism $\unicode[STIX]{x1D6FD}^{\prime }:\mathbb{Z}^{\unicode[STIX]{x1D6F4}(1)}\rightarrow N$ defined by

$$\begin{eqnarray}\unicode[STIX]{x1D6FD}^{\prime }(\unicode[STIX]{x1D70C})=\left\{\begin{array}{@{}ll@{}}\boldsymbol{d}(\unicode[STIX]{x1D70C})\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D70C}) & \text{if }\unicode[STIX]{x1D70C}\in \boldsymbol{\unicode[STIX]{x1D70C}},\\ \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D70C}) & \text{otherwise.}\end{array}\right.\end{eqnarray}$$

We denote the orbifold fan given by the triple $(N,\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D6FD}^{\prime })$ by $\boldsymbol{\unicode[STIX]{x1D6F4}}_{\boldsymbol{d}^{-1}\boldsymbol{\unicode[STIX]{x1D70C}}}$ . The natural morphism $\boldsymbol{\unicode[STIX]{x1D6F4}}_{\boldsymbol{d}^{-1}\boldsymbol{\unicode[STIX]{x1D70C}}}\rightarrow \boldsymbol{\unicode[STIX]{x1D6F4}}$ of orbifold fans, which is the identity map on the underlying group $N$ , is called the root construction of $\boldsymbol{\unicode[STIX]{x1D6F4}}$ with respect to the rays in $\boldsymbol{\unicode[STIX]{x1D70C}}$ with weights $\boldsymbol{d}$ .

The terminology in the definition above is, of course, motivated by its relation to the root stack of the corresponding toric stacks. Using the same notation as in the definition above, we let $\unicode[STIX]{x1D70B}:X^{\prime }\rightarrow X$ be the morphism of toric orbifolds associated to the root fan $\boldsymbol{\unicode[STIX]{x1D6F4}}_{\boldsymbol{d}^{-1}\boldsymbol{\unicode[STIX]{x1D70C}}}\rightarrow \boldsymbol{\unicode[STIX]{x1D6F4}}$ . On both $X$ and $X^{\prime }$ we have toric divisors corresponding to the rays $\unicode[STIX]{x1D70C}_{1},\ldots ,\unicode[STIX]{x1D70C}_{r}$ . Denote the sets of such divisors by $\boldsymbol{D}=\{D_{1},\ldots ,D_{r}\}$ and $\boldsymbol{D}^{\prime }=\{D_{1}^{\prime },\ldots ,D_{r}^{\prime }\}$ , respectively. Then each divisor $D_{i}^{\prime }$ is a $d_{i}$ th root of $\unicode[STIX]{x1D70B}^{\ast }D_{i}$ , and this structure identifies $X^{\prime }\rightarrow X$ with the root stack $X_{\boldsymbol{d}^{-1}\boldsymbol{D}}\rightarrow X$ , where we consider $\boldsymbol{d}$ as a function on $\boldsymbol{D}$ in the obvious way.

In terms of homogeneous coordinates, the root stack of a basic toric stack has the following description. Let $X$ be a basic toric stack with homogeneous coordinates $({\mathcal{O}}_{S}[x_{1},\ldots ,x_{r}],A,\boldsymbol{a})$ . Assume that $\boldsymbol{D}$ is a set of toric divisors corresponding to the coordinates $x_{1},\ldots ,x_{s}$ for some $s\leqslant r$ . Denote the generators of the group $\mathbb{Z}^{s}$ by $e_{1},\ldots ,e_{s}$ and define the group

$$\begin{eqnarray}A_{\boldsymbol{d}^{-1}\boldsymbol{a}}=A\oplus \mathbb{Z}^{s}/\langle d_{1}e_{1}-a_{1},\ldots ,d_{s}e_{s}-a_{s}\rangle ,\end{eqnarray}$$

which we think of as the group obtained from $A$ by formally adjoining the roots $e_{i}=a_{i}/d_{i}$ . Also, let $\boldsymbol{a}^{\prime }=(e_{1},\ldots ,e_{s},a_{s+1},\ldots ,a_{r})$ . Then the homogeneous coordinates of $X_{\boldsymbol{d}^{-1}\boldsymbol{D}}$ are given by

$$\begin{eqnarray}({\mathcal{O}}_{S}[x_{1}^{1/d_{1}},\ldots ,x_{s}^{1/d_{s}},x_{s+1},\ldots ,x_{r}],A_{\boldsymbol{d}^{-1}\boldsymbol{a}},\boldsymbol{a}^{\prime })\end{eqnarray}$$

and the map $X_{\boldsymbol{d}^{-1}\boldsymbol{D}}\rightarrow X$ corresponds to the map of graded rings taking $x_{i}$ to $x_{i}=(x_{i}^{1/d_{i}})^{d_{i}}$ .

Remark 3.15. Any toric stack $X=X_{\boldsymbol{\unicode[STIX]{x1D6F4}}}$ can be built up from the corresponding toric variety $X_{\text{cs}}=X_{\unicode[STIX]{x1D6F4}}$ as follows. There are canonical morphisms

$$\begin{eqnarray}X\rightarrow X_{\text{rig}}\rightarrow X_{\text{can}}\rightarrow X_{\text{cs}},\end{eqnarray}$$

where the first is a gerbe (the rigidification of $X$ ), the second is a root stack and the third is the canonical stack associated to $X_{\unicode[STIX]{x1D6F4}}$ . We describe this in terms of basic toric stacks. Let $X$ be a basic toric stack with homogeneous coordinates $({\mathcal{O}}_{S}[x_{1},\ldots ,x_{r}],A,(a_{1},\ldots ,a_{r}))$ . Then its rigidification $X_{\text{rig}}$ is a basic toric orbifold with homogeneous coordinates $({\mathcal{O}}_{S}[x_{1},\ldots ,x_{r}],A_{\text{rig}},(a_{1},\ldots ,a_{r}))$ , with $A_{\text{rig}}=\langle a_{1},\ldots ,a_{r}\rangle$ (see Remark 3.5).

For each $i=1,\ldots ,r$ , we define $A_{i}=A_{\text{rig}}/\langle a_{1},\ldots ,\widehat{a}_{i},\ldots ,a_{r}\rangle$ , where $\widehat{a}_{i}$ indicates that the element $a_{i}$ is omitted. Then we have an exact sequence

$$\begin{eqnarray}0\rightarrow A_{\text{can}}\rightarrow A_{\text{rig}}\xrightarrow[{}]{\unicode[STIX]{x1D711}}A_{1}\times \cdots \times A_{r}\rightarrow 0,\end{eqnarray}$$

where $\unicode[STIX]{x1D711}$ is the morphism induced by the canonical projections $A\rightarrow A_{i}$ . Denote the order of $A_{i}$ by $d_{i}$ , and note that $a_{i}^{d_{i}}\in A_{\text{can}}$ for each $i=1,\ldots ,r$ . Furthermore, we have a graded homomorphism

$$\begin{eqnarray}({\mathcal{O}}_{S}[x_{1}^{d_{1}},\ldots ,x_{r}^{d_{1}}],A_{\text{can}},(a_{1}^{d_{1}},\ldots ,a_{r}^{d_{r}}))\rightarrow ({\mathcal{O}}_{S}[x_{1},\ldots ,x_{r}],A_{\text{rig}},(a_{1},\ldots ,a_{r}))\end{eqnarray}$$

given by obvious inclusion of sheaves of ${\mathcal{O}}_{S}$ -algebras. It is straightforward to verify that this corresponds to a $\boldsymbol{d}$ th root stack in the toric divisors, where $\boldsymbol{d}=(d_{1},\ldots ,d_{r})$ , and that the corresponding morphism $X_{\text{rig}}\rightarrow X_{\text{can}}$ of basic toric orbifolds is the canonical morphism to the canonical stack.

3.5 Multiplicity, independency and smoothness

The toric destackification algorithm, which is described in the next section, is based on the well-known toric desingularisation algorithm described in for instance [Reference Cox, Little and SchenckCLS11, § 11]. In particular, the multiplicity of a cone plays an important role. Here we will briefly recall the main properties of multiplicities. We will also introduce the related concept of independency of toric divisors.

As usual, we let $\boldsymbol{\unicode[STIX]{x1D6F4}}=(N,\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D6FD})$ be an orbifold fan and $\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6F4}$ a cone. Let $\unicode[STIX]{x1D70C}_{1},\ldots ,\unicode[STIX]{x1D70C}_{r}$ be the rays in $\unicode[STIX]{x1D70E}(1)$ , and let $u_{i}$ be the non-zero lattice point on the ray $\unicode[STIX]{x1D70C}_{i}$ which is closest to the origin. We associate the parallelotope

$$\begin{eqnarray}P_{\unicode[STIX]{x1D70E}}=\biggl\{\mathop{\sum }_{i=1}^{r}\unicode[STIX]{x1D706}_{i}u_{i}\biggm\vert0\leqslant \unicode[STIX]{x1D706}_{i}<1\biggr\}\end{eqnarray}$$

to the cone $\unicode[STIX]{x1D70E}$ . Then the number of lattice points in $P_{\unicode[STIX]{x1D70E}}$ is called the multiplicity of $\unicode[STIX]{x1D70E}$ and is denoted by $\text{mult}\,(\unicode[STIX]{x1D70E})$ . The multiplicity satisfies the basic property $\text{mult}\,(\unicode[STIX]{x1D70F})|\text{mult}\,(\unicode[STIX]{x1D70E})$ if $\unicode[STIX]{x1D70F}\preccurlyeq \unicode[STIX]{x1D70E}$ . It should be noted that the stacky structure $\unicode[STIX]{x1D6FD}$ plays no part in the definition of multiplicity. In particular, the multiplicity of a cone is preserved by the root construction. A cone $\unicode[STIX]{x1D70E}$ is called smooth provided that the multiplicity $\text{mult}\,(\unicode[STIX]{x1D70E})$ equals one. An orbifold fan is smooth provided that all its cones are smooth.

The multiplicity $\text{mult}\,(\unicode[STIX]{x1D709})$ at a point $\unicode[STIX]{x1D709}\in X_{\boldsymbol{\unicode[STIX]{x1D6F4}}}$ in the toric orbifold is defined as the multiplicity of the cone spanned by the rays corresponding to the toric divisors passing through $\unicode[STIX]{x1D709}$ . The multiplicity is one at $\unicode[STIX]{x1D709}$ if and only if the corresponding point in the coarse space $X_{\unicode[STIX]{x1D6F4}}$ is smooth over the base. In particular, the coarse space is smooth if and only if the orbifold fan is smooth.

The following proposition describes how to read off the multiplicity from the homogeneous coordinates of a basic toric orbifold.

Proposition 3.16. Let $X$ be a basic toric orbifold with homogeneous coordinates

$$\begin{eqnarray}({\mathcal{O}}_{S}[x_{1},\ldots ,x_{r}],A,\boldsymbol{a})\end{eqnarray}$$

and let $(N,\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D6FD})$ be an orbifold fan defining $X$ . Let $\unicode[STIX]{x1D70E}$ be the (unique) maximal cone in $\unicode[STIX]{x1D6F4}$ . Then $\text{mult}\,(\unicode[STIX]{x1D70E})=|A_{\text{can}}|$ , where $A_{\text{can}}$ is defined as in Remark 3.15.

Proof. Let $u_{1},\ldots ,u_{r}$ be the non-zero lattice points closest to the origin on the rays $\unicode[STIX]{x1D70C}_{1},\ldots ,\unicode[STIX]{x1D70C}_{r}\in \unicode[STIX]{x1D6F4}(1)$ . Let $(N,\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D6FD}_{\text{can}})$ be the orbifold fan describing the canonical stack. Then $\unicode[STIX]{x1D6FD}_{\text{can}}(\unicode[STIX]{x1D70C}_{i})=u_{i}$ . By [Reference Cox, Little and SchenckCLS11, (11.1.5)], the multiplicity of $\unicode[STIX]{x1D70E}$ is given by the index $[N:\mathbb{Z}u_{1}+\cdots +\mathbb{Z}u_{r}]$ . Hence, the result follows from the exact sequence

$$\begin{eqnarray}0\rightarrow M\xrightarrow[{}]{\unicode[STIX]{x1D6FD}_{\text{can}}^{\prime }}\text{Hom}_{\mathbb{ Z}}(\mathbb{Z}^{\unicode[STIX]{x1D6F4}(1)},\mathbb{Z})\rightarrow A_{\text{can}}\rightarrow 0,\quad \unicode[STIX]{x1D6FD}_{\text{can}}^{\prime }=\text{Hom}_{\mathbb{ Z}}(\unicode[STIX]{x1D6FD}_{\text{can}},\mathbb{Z})\end{eqnarray}$$

from the Cox construction for the canonical stack $X_{\text{can}}$ .◻

Singularities on toric varieties only occur at the intersections of the toric divisors. Generally singularities get worse at points where more divisors intersect in the sense that the multiplicity grows. We introduce the concept of independency to capture the idea of nice divisors which do not make singularities worse.

Proposition 3.19. Let $X$ be a basic toric orbifold with homogeneous coordinates

$$\begin{eqnarray}({\mathcal{O}}_{S}[x_{1},\ldots ,x_{r}],A,(a_{1},\ldots ,a_{r}))\end{eqnarray}$$

and let $(N,\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D6FD})$ be an orbifold fan defining $X$ . Denote the rays of $\unicode[STIX]{x1D6F4}$ by $\unicode[STIX]{x1D70C}_{1},\ldots ,\unicode[STIX]{x1D70C}_{r}$ and let $\unicode[STIX]{x1D70E}$ denote the (unique) maximal cone of  $\unicode[STIX]{x1D6F4}$ . Then $\unicode[STIX]{x1D70C}_{i}$ is independent at $\unicode[STIX]{x1D70E}$ if and only if the intersection $\langle a_{1},\ldots ,\widehat{a}_{i},\ldots ,a_{r}\rangle \cap \langle a_{i}\rangle$ is trivial.

Proof. We assume, without loss of generality, that $i=r$ . Let $\unicode[STIX]{x1D70F}$ be the cone spanned by the rays $\unicode[STIX]{x1D70C}_{1},\ldots ,\unicode[STIX]{x1D70C}_{r-1}$ . By Proposition 3.16, the open substack $U_{\unicode[STIX]{x1D70F}}\subset U_{\unicode[STIX]{x1D70E}}=X$ has homogeneous coordinates

$$\begin{eqnarray}({\mathcal{O}}_{S}[x_{1},\ldots ,x_{r}][x_{r}^{-1}],A^{\prime },(\bar{a}_{1},\ldots ,\bar{a}_{r})),\end{eqnarray}$$

where $A^{\prime }=A/\langle a_{r}\rangle$ . Define $A_{\text{can}}$ and $A_{i}$ as in Remark 3.15, and similarly for $A_{\text{can}}^{\prime }$ and $A_{i}^{\prime }$ . Since the morphism $\langle a_{r}\rangle \rightarrow A_{r}$ is surjective, the snake lemma gives us the diagram

with exact rows and columns. By Proposition 3.16, we have $\text{mult}\,(\unicode[STIX]{x1D70E})=|A_{\text{can}}|$ and $\text{mult}\,(\unicode[STIX]{x1D70F})=|A_{\text{can}}^{\prime }|$ . Thus, it follows from Definition 3.17 and the diagram above that the ray $\unicode[STIX]{x1D70C}_{r}$ is independent at $\unicode[STIX]{x1D70E}$ if and only if $K$ vanishes. This is obviously equivalent to the condition stated in the proposition.◻