Orbifold pseudo-effective cones of toric stacks

TL;DR

This paper explicitly characterizes the orbifold pseudo-effective cone of split toric stacks, providing a detailed geometric description relevant to algebraic geometry and stack theory.

Contribution

It offers a novel explicit description of the orbifold pseudo-effective cone specifically for split toric stacks, advancing understanding in this area.

Findings

Explicit description of the orbifold pseudo-effective cone

Application to split toric stacks

Enhanced geometric understanding

Abstract

In this paper, we explicitly describe the orbifold pseudo-effective cone of a split toric stack.

Equations

$$N_{\mathrm{orb}}^1(\mathcal{X})=N^1(\mathcal{X})\oplus \underset{\mathcal{Y}\in {\pi }_0^{\ast }({\mathcal{J}}_0\mathcal{X})}{⨁}\mathbb{R}{u}_{\mathcal{Y}},$$

$$[{\mathcal{E}}_{\rho }]-\sum _{\mathcal{Y}\in {\pi }_0^{\ast }({\mathcal{J}}_0\mathcal{X})}{a}_{\rho }(\mathcal{Y})u_{\mathcal{Y}}(\rho \in \Sigma (1))$$

$$u_{\mathcal{Y}}(\mathcal{Y}\in {\pi }_0^{\ast }({\mathcal{J}}_0\mathcal{X})).$$

$$\beta :\underset{\rho \in \Sigma (1)}{⨁}\mathbb{Z}{\mathbf{v}}_{\rho }\to N$$

$$\mathbf{y}=\sum _{\rho \in \Sigma (1)}{a}_{\rho }(\mathbf{y})b_{\rho }.$$

$${\mathrm{Box}}^{\mathrm{rig}}(\Sigma ):=\{\mathbf{y}\in N^{\mathrm{rig}}\mid \forall \rho \in \Sigma (1),a_{\rho }(\mathbf{y})<1\}.$$

$${\pi }_0({\mathcal{J}}_{\infty }\mathcal{X})=\mathrm{Box}(\Sigma )\text{ and }{\pi }_0({\mathcal{J}}_{\infty }{\mathcal{X}}^{\mathrm{rig}})={\mathrm{Box}}^{\mathrm{rig}}(\Sigma ).$$

$$q:N^{\mathrm{rig}}\to {\mathrm{Box}}^{\mathrm{rig}}(\Sigma ),\mathbf{y}=\sum _{\rho \in \Sigma (1)}{a}_{\rho }(\mathbf{y})b_{\rho }\mapsto \sum _{\rho \in \Sigma (1)}\{a_{\rho }(\mathbf{y})\}{b}_{\rho }.$$

$$q\times {\mathrm{id}}_{G^D}:N^{\mathrm{rig}}\times G^D\to \mathrm{Box}(\Sigma )={\mathrm{Box}}^{\mathrm{rig}}(\Sigma )\times G^D,$$

$$U=\underset{\rho \in \Sigma (1)}{⨁}\mathbb{R}{\mathbf{u}}_{\rho }\text{ and }V=\underset{\rho \in \Sigma (1)}{⨁}\mathbb{R}{\mathbf{v}}_{\rho }.$$

$$M:=\mathrm{Hom}(N,\mathbb{Z})=\mathrm{Hom}(N^{\mathrm{rig}},\mathbb{Z})\cong {\mathbb{Z}}^d$$

$$\alpha :M_{\mathbb{R}}\to U,\mathbf{m}\mapsto \sum _{\rho \in \Sigma (1)}⟨w_{\rho },\mathbf{m}⟩{\mathbf{u}}_{\rho }$$

$$\lambda :U\to N^1(X),{\mathbf{u}}_{\rho }\mapsto [E_{\rho }],$$

$$0\to M_{\mathbb{R}}\stackrel{\alpha }{\to }U\stackrel{\lambda }{\to }{N}^1(X)\to 0.$$

$$0\to N_1(X)\stackrel{\mu }{\to }V\stackrel{{\beta }^{\prime }}{\to }{N}_{\mathbb{R}}\to 0.$$

$$\mathrm{Ker}({\beta }^{\prime })=\left\{\sum _{\rho \in \Sigma (1)}{a}_{\rho }{\mathbf{v}}_{\rho }|\sum _{\rho \in \Sigma (1)}{a}_{\rho }{w}_{\rho }=0\right\}.$$

$$U_{\mathrm{orb}}=\underset{\eta \in \Sigma (1)\cup {\pi }_0^{\ast }({\mathcal{J}}_0\mathcal{X})}{⨁}\mathbb{R}{\mathbf{u}}_{\eta }\text{ and }{V}_{\mathrm{orb}}=\underset{\eta \in \Sigma (1)\cup {\pi }_0^{\ast }({\mathcal{J}}_0\mathcal{X})}{⨁}\mathbb{R}{\mathbf{v}}_{\eta }.$$

$$N_{\mathrm{orb}}^1(\mathcal{X}):=N^1(X)\oplus \underset{\mathcal{Y}\in {\pi }_0^{\ast }({\mathcal{J}}_0\mathcal{X})}{⨁}\mathbb{R}{u}_{\mathcal{Y}}$$

$$N_{1,\mathrm{orb}}(\mathcal{X}):=N_1(X)\oplus \underset{\mathcal{Y}\in {\pi }_0^{\ast }({\mathcal{J}}_0\mathcal{X})}{⨁}\mathbb{R}{v}_{\mathcal{Y}}.$$

$$0\to 0\to \underset{\mathcal{Y}\in {\pi }_0^{\ast }({\mathcal{J}}_0\mathcal{X})}{⨁}\mathbb{R}{\mathbf{u}}_{\mathcal{Y}}\stackrel{{\mathbf{u}}_{\mathcal{Y}}\mapsto u_{{}^{\mathcal{Y}}}}{\to }\underset{\mathcal{Y}\in {\pi }_0^{\ast }({\mathcal{J}}_0\mathcal{X})}{⨁}\mathbb{R}{u}_{\mathcal{Y}}\to 0$$

$$0\to M_{\mathbb{R}}\stackrel{{\alpha }_{\mathrm{orb}}}{\to }{U}_{\mathrm{orb}}\stackrel{{\lambda }_{\mathrm{orb}}}{\to }{N}^1(\mathcal{X}{)}_{\mathrm{orb}}\to 0.$$

$$0\to N_{1,\mathrm{orb}}(\mathcal{X})\stackrel{{\mu }_{\mathrm{orb}}}{\to }{V}_{\mathrm{orb}}\stackrel{{\beta }_{\mathrm{orb}}^{\prime }}{\to }{N}_{\mathbb{R}}\to 0.$$

$$\mathrm{Ker}({\beta }_{\mathrm{orb}}^{\prime })$$

$$=\mathrm{Ker}({\beta }^{\prime })\oplus \underset{\mathcal{Y}\in {\pi }_0^{\ast }({\mathcal{J}}_0\mathcal{X})}{⨁}\mathbb{R}{\mathbf{v}}_{\mathcal{Y}}.$$

$$D^{\ast }\hookrightarrow {\mathcal{D}}^l\to \mathcal{X}.$$

$$({\mathcal{J}}_{\infty }\mathcal{X})⟨k⟩\to {\pi }_0({\mathcal{J}}_0\mathcal{X}),\gamma \to {\mathcal{Y}}_{\gamma }$$

$${\log }_{\mathcal{T}}:\mathcal{T}⟨k\text{llparenthesis}t\text{rrparenthesis}⟩\to N.$$

$$(k\text{llparenthesis}t{\text{rrparenthesis}}^{\ast }{)}^d\times \prod _{i=1}^{s}k\text{llparenthesis}t{\text{rrparenthesis}}^{\ast }/(k\text{llparenthesis}t{\text{rrparenthesis}}^{\ast }{)}^{l_i}$$

$$((h_i{)}_{1\le i\le d},([g_i]{)}_{1\le i\le s})$$

$$q\circ {\log }_{\mathcal{T}}:\mathcal{T}⟨k\text{llparenthesis}t\text{rrparenthesis}⟩\to N^{\mathrm{rig}}\times G^D\stackrel{q}{\to }{\mathrm{Box}}^{\mathrm{rig}}(\Sigma )\times G^D=\mathrm{Box}(\Sigma ).$$

$$\gamma :{\mathcal{D}}^l\stackrel{\sim }{\to }{\mathcal{C}}_z\to \mathcal{X}$$

Full text

1. Introduction

In [DY24], we introduced the notion of the orbifold pseudo-effective cones in order to generalize the Batyrev–Manin conjecture to Deligne–Mumford stacks. For varieties, this notion reduces to the notion of usual pseudo-effective cones. In [DY23], we verified the conjecture in the case where the stack in question is a split toric stack and the height function is the one associated to the anti-canonical raised line bundle. In the same paper, we presented a candidate description of the orbifold pseudo-effective cone of a split toric stack [DY23, Remark 3.4.2], however we could not verify that the description was correct. The aim of this paper is to prove that it is indeed the correct description of the orbifold pseudo-effective cone.

To state our main result more precisely, let us first recall how we can describe the pseudo-effective cone of a complete smooth toric variety. In what follows, we work over a field $k$ of characteristic zero. Let $\Sigma$ be a complete smooth fan in $N_{\mathbb{R}}=N\otimes\mathbb{R}$, where $N$ is a free abelian group of finite rank. Let $X$ be the toric variety associated to $\Sigma$ over $k$ and let $E_{\rho}$ be the torus-invariant prime divisors on $X$ corresponding to one-dimensional cones $\rho\in\Sigma(1)$ respectively, where $\Sigma(1)$ denotes the set of one-dimensional cones in $\Sigma$. The pseudo-effective cone $\overline{\mathrm{Eff}}(X)$ of $X$, which is a cone in the Néron–Severi space $\operatorname{\mathrm{N}}^{1}(X)$, is the cone generated by the numerical classes $[E_{\rho}]$, $\rho\in\Sigma(1)$ of torus-invariant prime divisors.

Next, consider a stacky fan $\mathbf{\Sigma}=(\Sigma,N,\beta)$ in the sense of Borisov–Chen–Smith [BCS04] and the associated toric stack $\mathcal{X}$ over $k$. We again assume that the fan $\Sigma$ is complete. So, $\mathcal{X}$ is a proper and smooth Deligne–Mumford stack over $k$. Note that $\mathcal{X}$ is a so-called split toric stack, namely the stacky torus contained in $\mathcal{X}$ as an open substack is isomorphic to $(\mathbb{G}_{m})^{d}\times\prod_{i=1}^{s}\operatorname{\mathrm{B}}\mu_{l_{i}}$ over $k$, where $d$ is the dimension of $\mathcal{X}$ and $l_{1},\dots,l_{s}$ are positive integers. The orbifold pseudo-effective cone of $\mathcal{X}$, denoted by $\overline{\mathrm{Eff}}_{\mathrm{orb}}(\mathcal{X})$, is a cone in the orbifold Néron–Severi space of $\mathcal{X}$, a modification of the usual Néron–Severi space by incorporating twisted sectors. Let $\pi_{0}^{*}(\mathcal{J}_{0}\mathcal{X})$ be the finite set of twisted sectors of $\mathcal{X}$. The orbifold Néron–Severi space is defined as

[TABLE]

where $\operatorname{\mathrm{N}}^{1}(\mathcal{X})$ is the usual Néron–Severi space and $u_{\mathcal{Y}}$ are indeterminants corresponding to twisted sectors $\mathcal{Y}$ respectively. Note that there is a one-to-one correspondence between twisted sectors and nonzero elements of a certain subset of the finitely generated abelian group $N$ (an input of the stacky fan $\mathbf{\Sigma}$) called the box. Using this correspondence and the structure of the stacky fan, we can associate a non-negative rational number $a_{\rho}(\mathcal{Y})$ to each pair $(\rho,\mathcal{Y})$ of $\rho\in\Sigma(1)$ and $\mathcal{Y}\in\pi_{0}^{*}(\mathcal{J}_{0}\mathcal{X})$ in an explicit way. The main result of this paper is stated as follows:

Theorem 1.1** (Corollary 7.8).**

Let $\mathcal{E}_{\rho}\subset\mathcal{X}$ be the torus-invariant prime divisors on $\mathcal{X}$ corresponding to $\rho\in\Sigma(1)$ respectively and let $[\mathcal{E}_{\rho}]$ denote their numerical classes. Then, the orbifold pseudo-effective cone $\overline{\mathrm{Eff}}_{\mathrm{orb}}(\mathcal{X})$ of $\mathcal{X}$ is generated by the $\#(\Sigma(1)\cup\pi_{0}^{*}(\mathcal{J}_{0}\mathcal{X}))$ elements

[TABLE]

and

[TABLE]

Example 1.2**.**

Let $\mathcal{X}$ be the weighted projective stack $\mathcal{P}(2,1)$. This is the projective line given with a stacky point at $\infty$ with automorphism group $\mu_{2}$. Thus, there exists a unique twisted sector $\mathcal{Y}$, lying over $\infty$. The corresponding fan has two one-dimensional rays $\rho_{0}$ and $\rho_{\infty}$ that corresponds to the points [math] and $\infty$, respectively. The orbifold Néron–Severi space $\operatorname{\mathrm{N}}_{\mathrm{orb}}^{1}(\mathcal{X})$ is of dimension two and has a basis $[\mathcal{E}_{\rho_{0}}]=2[\mathcal{E}_{\rho_{\infty}}]$, $u_{\mathcal{Y}}$. Theorem 1.1 says that the orbifold pseudo-effective cone $\overline{\mathrm{Eff}}_{\mathrm{orb}}(\mathcal{X})$ is generated by the three elements $[\mathcal{E}_{\rho_{0}}]$, $[\mathcal{E}_{\rho_{\infty}}]-\frac{1}{2}u_{\mathcal{Y}}$, and $u_{\mathcal{Y}}$. But, the first element $[\mathcal{E}_{\rho_{0}}]$ is obviously redundant and $\overline{\mathrm{Eff}}_{\mathrm{orb}}(\mathcal{X})$ is, in fact, generated by $[\mathcal{E}_{\rho_{\infty}}]-\frac{1}{2}u_{\mathcal{Y}}$, and $u_{\mathcal{Y}}$.

Our strategy of the proof is to describe the dual cone of $\overline{\mathrm{Eff}}_{\mathrm{orb}}(\mathcal{X})$, denoted by $\overline{\mathrm{Mov}}_{1,\mathrm{orb}}(\mathcal{X})$. It is defined to be the closure of the cone generated by the orbifold numerical classes of covering families of stacky curves. A difficult part in getting a desired description of $\overline{\mathrm{Mov}}_{1,\mathrm{orb}}(\mathcal{X})$ is to construct a covering family of stacky curves that induces a prescribed orbifold numerical class. We will do this by adapting Payne’s argument in [Pay06] for our stacky setting; we construct a stacky curve having a desired numerical property by combining several one-parameter subgroups in an appropriate way.

Note that Theorem 1.1 holds over an arbitrary field of characteristic zero. However, we may assume that the base field is algebraically closed, since the action of the absolute Galois group of $k$ on $\operatorname{\mathrm{N}}_{\mathrm{orb}}^{1}(\mathcal{X})$ is trivial (since $\mathcal{X}$ is a split toric stack) and hence the orbifold pseudo-effective cone does not change by passing to an algebraic closure of the base field. For this reason, we will assume that the base field $k$ is algebraically closed in the text of the paper. This simplifies our notation, since we can then focus on $k$-points of schemes and stacks at most parts of arguments rather than treating $L$-points for various extensions $L/k$.

Acknowledgements

We would like to thank Sho Tanimoto for helpful discussions. Ratko Darda has received a funding from the European Union’s Horizon 2023 research and innovation programme under the Maria Skłodowska-Curie grant agreement 101149785. Takehiko Yasuda was supported by JSPS KAKENHI Grant Numbers JP21H04994, JP23H01070 and JP24K00519.

2. Toric stacks and stacky fans

From now on, we assume that $k$ is an algebraically closed field of characteristic zero. However, we note again that Theorem 1.1 holds over any field of characteristic zero, as explained in Introduction.

We recall the notation on toric stacks from [DY23] with some modifications. There are several versions of toric stacks in the literature. We use the version introduced by Borisov–Chen–Smith [BCS04]. Let $N$ be a finitely generated abelian group of rank $d$. We fix a decomposition $N=N^{\mathrm{rig}}\oplus N_{\mathrm{tor}}$ of $N$ into a free abelian group $N^{\mathrm{rig}}$ of rank $d$ and a torsion group $N_{\mathrm{tor}}$. We also fix an isomorphism $N_{\mathrm{tor}}=\prod_{i=1}^{s}\mathbb{Z}/l_{i}\mathbb{Z}$. Let $N_{\mathbb{R}}:=N\otimes\mathbb{R}$, which is a $\mathbb{R}$-vector space of dimension $d$ and naturally identified with $N_{\mathbb{R}}^{\mathrm{rig}}:=N^{\mathrm{rig}}\otimes\mathbb{R}$.

Let $\Sigma$ be a complete simplicial fan in $N_{\mathbb{R}}$, which corresponds to a complete, normal and $\mathbb{Q}$-factorial toric variety $X=X_{\Sigma}$ of dimension $d$ over the base field $k$. Let $\bigoplus_{\rho\in\Sigma(1)}\mathbb{Z}\mathbf{v}_{\rho}$ be a free abelian group of rank $\#\Sigma(1)$ with $\Sigma(1)$ denoting the set of 1-dimensional cones in $\Sigma$ and let

[TABLE]

be a homomorphism such that the triple $\mathbf{\Sigma}=(\Sigma,N,\beta)$ is a stacky fan in the sense of [BCS04]. Let $\mathcal{X}=\mathcal{X}_{\mathbf{\Sigma}}$ be the toric stack corresponding to the stacky fan $\mathbf{\Sigma}$. This is a smooth and proper DM stack of dimension $d$ over $k$. There exists a natural morphism $\mathcal{X}\to X$, which makes $X$ a coarse moduli space of $\mathcal{X}$.

Let $\beta^{\mathrm{rig}}$ denote the composition of $\beta$ with the quotient map $N\to N^{\mathrm{rig}}$. The triple $\mathbf{\Sigma}^{\mathrm{rig}}:=(\Sigma,N^{\mathrm{rig}},\beta^{\mathrm{rig}})$ is again a stacky fan. The associated toric stack $\mathcal{X}^{\mathrm{rig}}:=\mathcal{X}_{\mathbf{\Sigma}^{\mathrm{rig}}}$ has a smooth and proper DM stack which has trivial generic stabilizer. There exists a natural morphism $\mathcal{X}\to\mathcal{X}^{\mathrm{rig}}$, which is the rigidification of $\mathcal{X}$ with respect to the dominating components of the inertia stack $\operatorname{\mathrm{I}}\mathcal{X}$. We also have a natural morphism $\mathcal{X}^{\mathrm{rig}}\to X$, which allows us to regard $X$ also as a coarse moduli space of $\mathcal{X}^{\mathrm{rig}}$. The composition $\mathcal{X}\to\mathcal{X}^{\mathrm{rig}}\to X$ is identical to the morphism $\mathcal{X}\to X$ above.

Let $T\cong\mathbb{G}_{m}^{d}$ be the algebraic torus corresponding to $N^{\mathrm{rig}}$ and let $\mathcal{T}=T\times\prod_{i=1}^{s}\operatorname{\mathrm{B}}\mu_{l_{i}}$ be the stacky torus corresponding to $N$. The toric stack $\mathcal{X}$ contains $\mathcal{T}$ as an open substack. From [FMN10, Definition 3.1 and Theorem 7.24], we have an action of $\mathcal{T}$ on $\mathcal{X}$.

For each $\rho\in\Sigma(1)$, let $b_{\rho}:=\beta^{\mathrm{rig}}(\mathbf{v}_{\rho})\in N^{\mathrm{rig}}$ and let $w_{\rho}$ be the primitive element of $N^{\mathrm{rig}}\cap\rho$, where we identify $N^{\mathrm{rig}}$ with a subgroup of $N_{\mathbb{R}}$ in the obvious way. There exists a unique $c_{\rho}\in\mathbb{Z}_{>0}$ with $b_{\rho}=c_{\rho}w_{\rho}$. Let $\mathcal{E}_{\rho},\mathcal{E}_{\rho}^{\mathrm{rig}},E_{\rho}$ be the prime divisors on $\mathcal{X}$, $\mathcal{X}^{\mathrm{rig}}$ and $X$ respectively that correspond to $\rho$. In particular, they are irreducible and reduced closed substacks of $\mathcal{X}$, $\mathcal{X}^{\mathrm{rig}}$ and $X$ respectively and there are natural morphisms $\mathcal{E}_{\rho}\to\mathcal{E}_{\rho}^{\mathrm{rig}}\to E_{\rho}$. The pullbacks of $E_{\rho}$ to $\mathcal{X}^{\mathrm{rig}}$ and $\mathcal{X}$ are $c_{\rho}\mathcal{E}_{\rho}$ and $c_{\rho}\mathcal{E}_{\rho}^{\mathrm{rig}}$.

Remark 2.1*.*

We should not confuse $\mathcal{E}_{\rho}^{\mathrm{rig}}$ with the rigidification of $\mathcal{E}_{\rho}$ with respect to dominant components of its inertia stack. Since the stack $\mathcal{E}_{\rho}^{\mathrm{rig}}$ can have non-trivial generic stabilizer, it is only the rigidification of $\mathcal{E}_{\rho}$ with respect to *some *dominant components of the inertia stack.

3. Sectors and boxes

The stack of twisted 0-jets of $\mathcal{X}$, denoted by $\mathcal{J}_{0}\mathcal{X}$, is the stack parametrizing representable morphisms $\operatorname{\mathrm{B}}\mu_{l}\to\mathcal{X}$, $l\in\mathbb{Z}_{>0}$. It is again a smooth and proper Deligne–Mumford stack over $k$. The original $\mathcal{X}$ is regarded as a connected component of $\mathcal{J}_{0}\mathcal{X}$. Let $\pi_{0}(\mathcal{J}_{0}\mathcal{X})$ be the set of connected component of $\mathcal{J}_{0}\mathcal{X}$ and let $\pi_{0}^{*}(\mathcal{J}_{0}\mathcal{X}):=\pi_{0}(\mathcal{J}_{0}\mathcal{X})\setminus\{\mathcal{X}\}$. These are finite sets. An element of $\pi_{0}(\mathcal{J}_{0}\mathcal{X})$ (resp. $\pi_{0}^{*}(\mathcal{J}_{0}\mathcal{X})$) is called a sector (resp. a twisted sector) of $\mathcal{X}$. See [DY24, Section 2.1] for more details.

For $\boldsymbol{y}\in N^{\mathrm{rig}}$, there are rational numbers $a_{\rho}(\boldsymbol{y})\in\mathbb{Q}_{\geq 0}$, $\rho\in\Sigma(1)$ such that $a_{\rho}(\boldsymbol{y})>0$ if and only if $\rho$ is a face of the minimal cone $\sigma\in\Sigma$ containing $\boldsymbol{y}$ and such that we can write

[TABLE]

See [DY23, Section 3.1.1]. We define

[TABLE]

We define $\mathrm{Box}(\Sigma)$ to be the preimage of $\mathrm{Box}^{\mathrm{rig}}(\Sigma)$ by the projection $N=N^{\mathrm{rig}}\oplus N_{\mathrm{tor}}\to N^{\mathrm{rig}}$, which is identified with $\mathrm{Box}^{\mathrm{rig}}(\Sigma)\times N_{\mathrm{tor}}$ in the obvious way. There are canonical identifications

[TABLE]

See [BCS04, Proposition 4.7] and [DY23, Lemma 3.1.4].

We have a map

[TABLE]

Here, for a rational number $s$, $\{s\}$ denotes the fractional part $s-\lfloor s\rfloor$ of $s$. We extend this map to

[TABLE]

which we denote again by $q$.

4. Exact sequences

4.1. “Non-orbifold” exact sequences

We first recall two standard exact sequences which are dual to each other and involve the spaces of divisors and 1-cycles, $M_{\mathbb{R}}$ and $N_{\mathbb{R}}$. Consider the following two $\mathbb{R}$-vector spaces

[TABLE]

We regard $U$ and $V$ as dual spaces to each other by the pairing determined by $\langle\mathbf{u}_{\rho},\mathbf{v}_{\rho^{\prime}}\rangle=\delta_{\rho,\rho^{\prime}}$ (Kronecker’s delta). Let

[TABLE]

and let $M_{\mathbb{R}}:=M\otimes\mathbb{R}\cong\mathbb{R}^{d}.$ Let $\operatorname{\mathrm{N}}^{1}(X)$ denote the Néron-Severi space of $X$ defined over $\mathbb{R}$ and let $\operatorname{\mathrm{N}}_{1}(X)$ be the space of numerical classes of $1$-cycles defined over $\mathbb{R}$. These are $\mathbb{R}$-vectors spaces dual to each other and their dimension is the Picard number of $X$.

We consider the map

[TABLE]

and the map

[TABLE]

where $[E_{\rho}]$ is the numerical class of $E_{\rho}$. As is well-known (for example, see [CLS11, Propositions 6.3.15 and 6.4.1]), these maps fit into the short exact sequence:

[TABLE]

Let us consider its dual exact sequence,

[TABLE]

Here, the map $\mu$ is given by $\mu(h)=\sum_{\rho\in\Sigma(1)}(\mu,E_{\rho})\mathbf{v}_{\rho}$ and the map $\beta^{\prime}$ is given by $\beta^{\prime}(\mathbf{v}_{\rho})=w_{\rho}$. Thus, $\operatorname{\mathrm{N}}_{1}(X)$ is identified with $\operatorname{Ker}(\beta^{\prime})$, which is explicitly described as

[TABLE]

Remark 4.1*.*

Let us denote the map $V\to N_{\mathbb{R}}$ obtained by tensoring $\beta$ with $\mathbb{R}$ again by $\beta$ and let $c\colon V\to V$ be the linear map with $\mathbf{v}_{\rho}\mapsto c_{\rho}\mathbf{v}_{\rho}$. Then, $\beta^{\prime}$ and $\beta$ is related by $\beta=c\circ\beta^{\prime}$.

4.2. “Orbifold” exact sequences

In what follows, we often treat elements of $\Sigma(1)$ and $\pi_{0}^{*}(\mathcal{J}_{0}\mathcal{X})$ equally by denoting them by $\eta$. On the other hand, when we treat them separately, we denote elements of $\Sigma(1)$ by $\rho$ and elements of $\pi_{0}^{*}(\mathcal{J}_{0}\mathcal{X})$ by $\mathcal{Y}$.

Next we construct the orbifold versions of the exact sequences (4.1) and (4.2) obtained above. Consider the following two $\mathbb{R}$-vector spaces

[TABLE]

We regard $U_{\mathrm{orb}}$ and $V_{\mathrm{orb}}$ as dual spaces to each other by the pairing $\langle\mathbf{u}_{\eta},\mathbf{v}_{\eta^{\prime}}\rangle=\delta_{\eta,\eta^{\prime}}$. Let

[TABLE]

be the orbifold Néron–Severi space defined over $\mathbb{R}$. Similarly, we define

[TABLE]

The intersection pairing between $\operatorname{\mathrm{N}}_{\mathrm{orb}}^{1}(\mathcal{X})$ and $\operatorname{\mathrm{N}}_{1,\mathrm{orb}}(\mathcal{X})$ is defined as the component-wise pairing. It makes them dual to each other. Taking the direct sum of (4.1) and

[TABLE]

gives the short exact sequence

[TABLE]

As the dual of the last sequence, we get the short exact sequence

[TABLE]

The map $\beta^{\prime}_{\mathrm{orb}}\colon V_{\mathrm{orb}}\to N_{\mathbb{R}}$ sends $\mathbf{v}_{\rho}$ to $w_{\rho}$ the primitive lattice vector in $N^{\mathrm{rig}}$ on $\rho$ and $\mathbf{v}_{\mathcal{Y}}$ to [math]. Thus, $\operatorname{\mathrm{N}}_{1,\mathrm{orb}}(\mathcal{X})$ is identified with the kernel of $\beta^{\prime}_{\mathrm{orb}}$, which is described as

[TABLE]

5. Orbifold numerical classes of stacky curves

5.1. $k\llparenthesis t\rrparenthesis$-points and twisted arcs

Let $D=\operatorname{Spec}k\llbracket t\rrbracket$ be the formal disk. For $l\in\mathbb{Z}_{>0}$, let $\mathcal{D}^{l}:=[\operatorname{Spec}k\llbracket t^{1/l}\rrbracket/\mu_{l}]$, a twisted formal disk. There exists a natural morphism $\mathcal{D}^{l}\to D$ making $D$ the coarse moduli space of $\mathcal{D}^{l}$. This morphism is an isomorphism over the punctured formal disk $D^{*}:=\operatorname{Spec}k\llparenthesis t\rrparenthesis\subset D$. Thus, we can regard $D^{*}$ also as an open substack of $\mathcal{D}^{l}$. A twisted arc of $\mathcal{X}$ is a representable morphism of the form $\mathcal{D}^{l}\to\mathcal{X}$ for some $l\in\mathbb{Z}_{>0}$. A twisted arc induces a $k\llparenthesis t\rrparenthesis$-point of $\mathcal{X}$ as the composition

[TABLE]

Conversely, for every $k\llparenthesis t\rrparenthesis$-point $\gamma\colon D^{*}\to\mathcal{X}$, there exists a unique twisted arc $\mathcal{D}^{l}\to\mathcal{X}$ inducing $\gamma$. Thus, we have a one-to-one correspondence between $\mathcal{X}\langle k\llparenthesis t\rrparenthesis\rangle$ and $(\mathcal{J}_{\infty}\mathcal{X})\langle k\rangle$. Here the notation $\mathcal{X}\langle L\rangle$ means the set of isomorphism classes in the groupoid $\mathcal{X}(L)$ of $L$-points of the stack $\mathcal{X}$.

In what follows, we often identify a $k\llparenthesis t\rrparenthesis$-point with its corresponding twisted arc and denote them by the same symbol, which is often $\gamma$.

5.2. The residue map

Definition 5.1**.**

Given a twisted arc $\gamma\colon\mathcal{D}^{l}\to\mathcal{X}$, composing it with the natural closed immersion $\operatorname{\mathrm{B}}\mu_{l}\hookrightarrow\mathcal{D}^{l}$, we get a $k$-point of $\mathcal{J}_{0}\mathcal{X}$. We define the* sector associated to* $\gamma$ to be the connected component of $\mathcal{J}_{0}\mathcal{X}$ containing this $k$-point and denote it by $\mathcal{Y}_{\gamma}$. We also call it the sector associated to $\gamma$. The map

[TABLE]

is called the residue map.

From [DY23, §2.1], we have the map

[TABLE]

Via the identification $\mathcal{T}=\mathbb{G}_{m}^{d}\times\prod_{i=1}^{s}\operatorname{\mathrm{B}}\mu_{l_{i}}$, the above map is identified with

[TABLE]

From [DY23, Lemmas 3.1.6 and 3.1.9], the residue map $\psi\colon\mathcal{T}\langle k\llparenthesis t\rrparenthesis\rangle\to\pi_{0}(\mathcal{J}_{0}\mathcal{X})$ is identical to

[TABLE]

5.3. Orbifold numerical classes

Definition 5.2**.**

By a *stacky curve *on $\mathcal{X}$, we mean a representable morphism $f\colon\mathcal{C}\to\mathcal{X}$ such that $\mathcal{C}$ is an irreducible smooth and proper DM stack over $k$ having the trivial generic stabilizer.

Let $f\colon\mathcal{C}\to\mathcal{X}$ be a stacky curve. For each $k$-point $z\in\mathcal{C}$, the formal completion $\mathcal{C}_{z}$ of $\mathcal{C}$ at $z$ is isomorphic to $\mathcal{D}^{l}:=[\operatorname{Spec}k\llbracket t^{1/l}\rrbracket/\mu_{l}]$. The composition

[TABLE]

is a twisted arc. The sector $\mathcal{Y}_{\gamma}$ associated to $\gamma$ is independent of the chosen isomorphism and depends only on $f$ and $z$. We call it the *sector of $f$ at $z$ *and denote it by $\mathcal{Y}_{f,z}$ or just by $\mathcal{Y}_{z}$.

Definition 5.3** ([DY24, Remark 8.9]).**

For a stacky curve $f\colon\mathcal{C}\to\mathcal{X}$, we define its orbifold numerical class to be

[TABLE]

where $\overline{f}$ is the morphism $\overline{\mathcal{C}}\to X$ derived from $f$ with $\overline{\mathcal{C}}$ the coarse moduli space of $\mathcal{C}$, $\mathcal{Y}_{z}$ is the twisted sector associated to $f$ and $x$.

5.4. Localization of orbifold numerical classes

We will see that when a stacky curve $f\colon\mathcal{C}\to\mathcal{X}$ maps the generic point into $\mathcal{T}$, then the orbifold numerical class $[f]_{\mathrm{orb}}$ is written as the summation of “local classes” $[f_{z}]_{\mathrm{orb}}$, $z\in\mathcal{C}\langle k\rangle$.

For $\rho\in\Sigma(1)$, let $\mathcal{I}_{\rho}\subset\mathcal{O}_{\mathcal{X}}$ be the coherent ideal sheaf defining the prime divisor $\mathcal{E}_{\rho}\subset\mathcal{X}$. For a twisted arc $\gamma\colon\mathcal{D}^{l}\to\mathcal{X}$, consider the composite morphism

[TABLE]

The pull-back $(\widetilde{\gamma})^{-1}\mathcal{I}_{\rho}\subset k\llbracket t^{1/l}\rrbracket$ is an ideal of the form $(t^{n/l})$, $n\in\mathbb{Z}_{\geq 0}\cup\{\infty\}$. Here we followed the convention $(t^{\infty})=(0)$. If the corresponding $k\llparenthesis t\rrparenthesis$-point lies in $\mathcal{T}$, then $n$ is finite.

Definition 5.4**.**

We define the intersection number of $\gamma$ and $\mathcal{E}_{\rho}$ to be

[TABLE]

with $n$ and $l$ as above. We define $(\gamma,E_{\rho}):=c_{\rho}(\gamma,\mathcal{E}_{\rho})$.

Definition 5.5**.**

For $\rho\in\Sigma(1)$, let $\mathbf{b}_{\rho}:=c_{\rho}\mathbf{v}_{\rho}\in V_{\mathrm{orb}}$. We define the *orbifold numerical class *of $\gamma\in\mathcal{T}(k\llparenthesis t\rrparenthesis)$ to be

[TABLE]

Proposition 5.6**.**

Let $f\colon\mathcal{C}\to\mathcal{X}$ be a stacky curve such that the generic point of $\mathcal{C}$ mapping into $\mathcal{T}$. Then, we have

[TABLE]

?proofname?

.

We first note that since $[f_{z}]_{\mathrm{orb}}=0$ all but finitely many points $z$, the sum in the proposition is a finite sum. We have

[TABLE]

∎

For $\gamma\in\mathcal{T}(k\llparenthesis t\rrparenthesis)$, let $u:=\log_{T}(\gamma)\in N^{\mathrm{rig}}$ and let $\sigma\in\Sigma$ be a $d$-dimensional cone containing $u$. The open substack $\mathcal{U}(\sigma)\subset\mathcal{X}$ corresponding to $\sigma$ is of the form $[\mathbb{A}_{k}^{d}/G_{\sigma}]$ for a finite abelian group $G_{\sigma}$. From the proof of [DY23, 3.1.9], the twisted arc $[\operatorname{Spec}k\llbracket t^{1/l}\rrbracket/\mu_{l}]\to\mathcal{X}$ that corresponds to $\gamma$ factors as

[TABLE]

For a unique embedding $\iota\colon\mu_{l}\hookrightarrow G_{\sigma}$, there exists a (non-unique) lift of this morphism to an $\iota$-equivariant morphism

[TABLE]

Let $\rho_{1},\dots,\rho_{d}$ be the one-dimensional faces of $\sigma$, let $\boldsymbol{y}\in\mathrm{Box}^{\mathrm{rig}}(\sigma)$ be the image of $u$ and let $a_{\rho_{i}}^{\sigma}(\boldsymbol{y})=a_{i}\in\frac{1}{l}\mathbb{Z}$. We have

[TABLE]

In particular,

[TABLE]

With the notation above, if $\rho_{1},\dots,\rho_{d}\in\Sigma(1)$ are such that $\mathcal{E}_{\rho_{i}}$, $1\leq i\leq d$, are defined by $x_{i}=0$ respectively, then

[TABLE]

For $\rho\in\Sigma(1)\setminus\{\rho_{1},\dots,\rho_{d}\}$, we have $(\gamma,\mathcal{E}_{\rho})=0$.

With the notation above, we can write

[TABLE]

Moreover, if $e$ is the image of $\gamma$ by the composite map

[TABLE]

then the sector $\mathcal{Y}_{\gamma}$ corresponds to the element

[TABLE]

Definition 5.7**.**

For each $\eta\in\Sigma(1)\cup\pi_{0}^{*}(\mathcal{J}_{0}\mathcal{X})$, we define an element $\Xi_{\eta}\in V_{\mathrm{orb}}$ as follows:

[TABLE]

For the non-twisted sector $\mathcal{X}\in\pi_{0}^{*}(\mathcal{J}_{0}\mathcal{X})$, we put $\Xi_{\mathcal{X}}:=0\in V_{\mathrm{orb}}$.

Lemma 5.8**.**

For $\gamma\in\mathcal{T}(k\llparenthesis t\rrparenthesis)$, we have

[TABLE]

?proofname?.

We have

[TABLE]

which is an element of $\Xi_{\mathcal{Y}_{\gamma}}+\sum_{\rho\in\Sigma(1)}\mathbb{Z}_{\geq 0}\Xi_{\rho}$. ∎

Corollary 5.9**.**

Let $f\colon\mathcal{C}\to\mathcal{X}$ be a stacky curve such that the generic point of $\mathcal{C}$ mapping into $\mathcal{T}$. Then, we have

[TABLE]

?proofname?.

This is a direct consequence of Lemma 5.8 and Proposition 5.6. ∎

5.5. The orbifold class derived by a one-parameter subgroup

For each $b\in N=N^{\mathrm{rig}}\times N_{\mathrm{tor}}$, we have the associated morphism

[TABLE]

For $\rho\in\Sigma(1)$, let $b_{\rho}$ denote $\beta(\mathbf{v}_{\rho})$. The natural map $k(t)\hookrightarrow k\llparenthesis t\rrparenthesis$ induces the $k\llparenthesis t\rrparenthesis$-point

[TABLE]

Lemma 5.10**.**

We have

[TABLE]

?proofname?.

By construction,

[TABLE]

with $g\in G^{D}$. Thus, the sector associated to $\widehat{\phi_{b}}$ is

[TABLE]

Thus,

[TABLE]

∎

Corollary 5.11**.**

For $\eta\in\Sigma(1)\cup\pi_{0}^{*}(\mathcal{J}_{0}\mathcal{X})$, we have $\left[\widehat{\phi_{b_{\eta}}}\right]_{\mathrm{orb}}=\Xi_{\eta}$.

?proofname?.

This is a direct consequence of the last lemma. ∎

6. The “action” of $\operatorname{\mathrm{J}}_{\infty}\mathcal{T}$ on $\mathcal{J}_{\infty}\mathcal{X}$

From [FMN10, Definition 3.1 and Theorem 7.24], we have an action

[TABLE]

of the stacky torus $\mathcal{T}$ on the toric stack $\mathcal{X}$.

Let $\operatorname{\mathrm{J}}_{\infty}\mathcal{T}$ denote the stack of (untwisted) arcs on $\mathcal{T}$. A $k$-point of $\operatorname{\mathrm{J}}_{\infty}\mathcal{T}$ corresponds to a morphism

[TABLE]

Let $\mathcal{J}_{\infty}\mathcal{X}$ denote the stack of twisted arcs on $\mathcal{X}$. A $k$-point of $\mathcal{J}_{\infty}\mathcal{X}$ corresponds to a representable morphism

[TABLE]

Let $\delta\colon D\to\mathcal{T}$ be an arc and let $\gamma\colon\mathcal{D}^{l}\to\mathcal{X}$ be a twisted arc. Let $\delta\cdot\gamma\colon\mathcal{D}^{l}\to\mathcal{X}$ be the twisted arc obtained as the composition

[TABLE]

Let $\mathcal{X}_{D}:=\mathcal{X}\times D$ and $\mathcal{T}_{D}:=\mathcal{T}\times D$. The arc $\delta$ also defines a $\mathrm{D}$-automorphism $\alpha_{\delta}\colon\mathcal{X}_{\mathrm{D}}\to\mathcal{X}_{\mathrm{D}}$. The twisted arc $\delta\cdot\gamma$ is isomorphic to $\alpha_{\delta}\circ\gamma$, which follows from the following 2-commutative diagram.

[TABLE]

To see that $\delta\cdot\gamma$ is indeed a twisted arc, we need to see that $\delta\cdot\gamma$ is a representable morphism. To show this, we take a quasi-inverse arc $\delta^{-1}\colon D\to\mathcal{T}$ of $\delta$. The induced morphism $\alpha_{\delta^{-1}}$ is a quasi-inverse to $\alpha_{\delta}$. This implies that $\alpha_{\delta}$ is representable, which shows that $\delta\cdot\gamma=\alpha_{\delta}\circ\gamma$ is also representable.

Lemma 6.1**.**

For every $\rho\in\Sigma(1)$, $(\delta\cdot\gamma,\mathcal{E}_{\rho})=(\gamma,\mathcal{E}_{\rho})$.

?proofname?.

Let $\mathcal{E}_{\rho,D}\subset\mathcal{X}_{D}$ be the closed substack derived from $\mathcal{E}_{\rho}$ by the base change. The isomorphism $\alpha_{\delta}$ preserves $\mathcal{E}_{\rho,D}$. Thus,

[TABLE]

This shows the lemma. ∎

Lemma 6.2**.**

The twisted arcs $\delta\cdot\gamma$ and $\gamma$ induce the same sector. Namely, $\mathcal{Y}_{\delta\cdot\gamma}=\mathcal{Y}_{\gamma}$.

?proofname?.

Let $\delta_{0}\colon\operatorname{Spec}k\to\mathcal{T}$ and $\gamma_{0}\colon\operatorname{\mathrm{B}}\mu_{l}\to\mathcal{X}$ be the morphisms obtained by restricting $\delta$ and $\gamma$ to the reduced closed substacks at the unique closed point of the source, respectively. We get an automorphism $\alpha_{\delta_{0}}\colon\mathcal{X}\to\mathcal{X}$ obtained from $\delta_{0}$. We need to show that the two morphisms $\alpha_{\delta_{0}}\circ\gamma_{0},\gamma_{0}\colon\operatorname{\mathrm{B}}\mu_{l}\to\mathcal{X}$ determines points in the same connected component of $\mathcal{J}_{0}\mathcal{X}$. Let $e\colon T\to\mathcal{T}$ be the canonical morphism. We get an automorphism $\alpha_{e}\colon\mathcal{X}_{T}\to\mathcal{X}_{T}$. This induces a morphism

[TABLE]

From the construction, the composition

[TABLE]

is the identity morphism. Therefore, for each sector $\mathcal{Y}\subset\mathcal{J}_{0}\mathcal{X}$, since $T$ is connected and $\mathcal{Y}$ is geometrically connected, the image of $T\times\mathcal{Y}$ is $\mathcal{Y}$ and get a morphism $T\times\mathcal{Y}\to\mathcal{Y}$. Let $\widetilde{\delta_{0}}\colon\operatorname{Spec}k\to T$ be a lift of $\delta_{0}$. If $\gamma_{0}$ corresponds to a point of $\mathcal{Y}$, then $\alpha_{\delta_{0}}\circ\gamma_{0}$ corresponds to

[TABLE]

In particular, this is a point of the same sector $\mathcal{Y}$, as desired. ∎

Corollary 6.3**.**

We have $[\delta\cdot\gamma]_{\mathrm{orb}}=[\gamma]_{\mathrm{orb}}$.

?proofname?.

This follows from Lemmas 6.1 and 6.2 and Definition 5.3. ∎

Remark 6.4*.*

We guess that our construction of $\delta\cdot\gamma$ would be eventually obtained from an action of $\operatorname{\mathrm{J}}_{\infty}\mathcal{T}$ on $\mathcal{J}_{\infty}\mathcal{X}$ in the sense of [FMN10, Appendix B]. In this paper, however, we have decided to content ourselves with showing only properties of $\delta\cdot\gamma$ that we need in the proof of the main result.

7. Orbifold pseudo-effective cones

Definition 7.1**.**

A covering family of stacky curves on $\mathcal{X}$ is the pair $(\pi\colon\widetilde{\mathcal{C}}\to S,\widetilde{f}\colon\widetilde{\mathcal{C}}\to\mathcal{X})$ of a smooth morphism $\pi\colon\widetilde{\mathcal{C}}\to S$ of DM stacks with $T$ an irreducible $k$-variety and a representable dominant morphism $\widetilde{f}\colon\widetilde{\mathcal{C}}\to\mathcal{X}$ such that for every $s\in S(k)$, the morphism

[TABLE]

is a stacky curve on $\mathcal{X}$.

Definition 7.2**.**

For a covering family $(\pi\colon\widetilde{\mathcal{C}}\to S,\widetilde{f}\colon\widetilde{\mathcal{C}}\to\mathcal{X})$ of stacky curves on $\mathcal{X}$, we define its *orbifold numerical class, *denoted by $[\widetilde{f}]_{\mathrm{orb}}$, to be $[\widetilde{f}|_{\pi^{-1}(s)}]$ for a general point $s\in S(k)$ (which is well-defined thanks to [DY24, Lemma 8.6]).

Let $f\colon\mathcal{C}\to\mathcal{X}$ be a stacky curve. From (4.3), $[f]_{\mathrm{orb}}$ lies in $\operatorname{Ker}(\beta_{\mathrm{orb}}^{\prime})$ and hence is regarded as an element of $\operatorname{\mathrm{N}}_{1,\mathrm{orb}}(\mathcal{X})$.

Definition 7.3**.**

We define a cone $\overline{\mathrm{Mov}}_{1,\mathrm{orb}}(\mathcal{X})\subset\operatorname{\mathrm{N}}_{1,\mathrm{orb}}(\mathcal{X})$ to be the closure of the cone generated by the classes $[\widetilde{f}]_{\mathrm{orb}}$ of all the covering families $(\pi\colon\widetilde{\mathcal{C}}\to S,\widetilde{f}\colon\widetilde{\mathcal{C}}\to\mathcal{X})$ of stacky curves on $\mathcal{X}$. We define the *orbifold pseudo-effective cone *$\overline{\mathrm{Eff}}_{\mathrm{orb}}(\mathcal{X})\subset\operatorname{\mathrm{N}}_{\mathrm{orb}}^{1}(\mathcal{X})$ to be the dual cone of $\overline{\mathrm{Mov}}_{1,\mathrm{orb}}(\mathcal{X})$.

Lemma 7.4**.**

The cone $\overline{\mathrm{Mov}}_{1,\mathrm{orb}}(\mathcal{X})\subset\operatorname{\mathrm{N}}_{1,\mathrm{orb}}(\mathcal{X})$ is the closure of the cone generated by by the classes $[f]_{\mathrm{orb}}$ of those stacky curves $f\colon\mathcal{C}\to\mathcal{X}$ that map the generic point into $\mathcal{T}$.

?proofname?.

It is enough to show that the two sets of orbifold numerical classes are identical; the orbifold numerical classes of covering families of stacky curves and the ones of stacky curves mapping the generic point into $\mathcal{T}$. If $(\pi\colon\widetilde{\mathcal{C}}\to S,\widetilde{f}\colon\widetilde{\mathcal{C}}\to\mathcal{X})$ is a covering family of stacky curves on $\mathcal{X}$, then for a general point $s\in S(k)$, $\widetilde{f}|_{\pi^{-1}(s)}$ is a stacky curve which maps the generic point into $\mathcal{T}$. Conversely, for a stacky curve $f\colon\mathcal{C}\to\mathcal{X}$ which maps the generic point into $\mathcal{T}$, we consider the composite morphism

[TABLE]

Here $T\times\mathcal{X}\to\mathcal{T}\times\mathcal{X}$ is the product of the standard atlas $T\to\mathcal{T}$ and the identity morphism $\mathrm{id}_{\mathcal{X}}$, and $\mathcal{T}\times\mathcal{X}\to\mathcal{X}$ is the action of $\mathcal{T}$ on $\mathcal{X}$. The pair of the projection $\pi\colon T\times\mathcal{C}\to T$ and $\widetilde{f}$ is a covering family. From Proposition 5.6 and Corollary 6.3, for every $t\in T(k)$, $[\widetilde{f}|_{\pi^{-1}(t)}]_{\mathrm{orb}}=[f]_{\mathrm{orb}}$, which completes the proof. ∎

Theorem 7.5**.**

The cone $\overline{\mathrm{Mov}}_{1,\mathrm{orb}}(\mathcal{X})$ is generated by the image $\overline{\Xi_{\eta}}$ of $\Xi_{\eta}$, $\eta\in\Sigma(1)\cup\pi_{0}^{*}(\mathcal{J}_{0}\mathcal{X})$:

[TABLE]

?proofname?.

Our proof is a variation of the proof of [Pay06, Proposition 2]. From Corollary 5.9, the left hand side is included in the right hand side. To show the opposite inclusion, we take an arbitrary element of

[TABLE]

where $\eta_{1},\dots,\eta_{r}$ is a sequence of elements of $\Sigma(1)\cup\pi_{0}^{*}(\mathcal{J}_{0}\mathcal{X})$. We will show that $\sum_{i=1}^{r}\Xi_{\eta_{i}}$ is an element of $\overline{\mathrm{Mov}}_{1,\mathrm{orb}}(\mathcal{X})$.

Let $\lambda_{1},\dots,\lambda_{r}\in k^{*}$ be distinct elements and consider the morphism

[TABLE]

Here the product is taken with respect to the group structure of $\mathcal{T}$. Let $f\colon\mathcal{C}\to\mathcal{X}$ be the stacky curve uniquely extending $\Phi$. We identify $\mathcal{C}\langle k\rangle$ with $\mathbb{P}^{1}\langle k\rangle=k\cup\{\infty\}$.

Claim 7.6*.*

The point $\infty\in\mathcal{C}\langle k\rangle$ is not a stacky point. Moreover, the image of $\infty$ by $\Phi$ in $\mathcal{T}\langle k\rangle=T\langle k\rangle$ is the identity element $1_{T}$ for the group structure of $T\langle k\rangle$.

Proof of Claim 7.6.

We first prove the second assertion. From (4.3), we have $\sum b_{\eta_{i}}=0$. Since $\Psi(z)=\prod\phi_{b_{\eta_{i}}}(z)$ is the trivial morphism $1\colon\mathbb{G}_{m}\to\mathcal{Y}$, we have

[TABLE]

as desired. Considering the affine line $\mathbb{A}^{1}$ with coordinate $w=1/z$, we have the morphism

[TABLE]

which corresponds to $\Phi$ via the coordinate change $w=1/z.$ This shows that the point $z=\infty$ (corresponding to $w=0$) is not a stacky point of $\mathcal{C}$. ∎

Claim 7.7*.*

For $i\in\{1,\dots,r\}$, we have $[\Phi_{\lambda_{i}}]_{\mathrm{orb}}=\Xi_{\eta_{i}}$.

Proof of Claim 7.7.

For the parameter $t:=z-\lambda_{i}$, $\phi_{b_{\eta_{i}}}$ induces the $k\llparenthesis t\rrparenthesis$-point $\widehat{\phi_{b_{\eta_{i}}}}$ of $\mathcal{T}$ with the induced twisted arc $\gamma_{i}\colon\mathcal{D}^{l}\to\mathcal{X}$ and $\phi_{b_{\eta_{j}}}$ with $j\neq i$ induce $k\llparenthesis t\rrparenthesis$-points of $\mathcal{T}$ which extend to arcs $\delta_{j}\colon D\to\mathcal{T}$. We have

[TABLE]

From Corollary 5.11, we have $\left[\gamma_{i}\right]_{\mathrm{orb}}=\Xi_{\eta_{i}}$. From Corollary 6.3, $[\Phi_{\lambda_{i}}]_{\mathrm{orb}}=\Xi_{\eta_{i}}$, as desired. ∎

Claims 7.6 and 7.7 show

[TABLE]

We have completed the proof of Theorem 7.5. ∎

Corollary 7.8**.**

For $\eta\in\Sigma(1)\cup\pi_{0}^{*}(\mathcal{J}_{0}\mathcal{X})$, let $\{\Xi_{\eta}\geq 0\}$ denote the half-space $\{\theta\in U_{\mathrm{orb}}\mid(\theta,\Xi_{\eta})\geq 0\}$ in $U_{\mathrm{orb}}$. Then, the orbifold pseudo-effective cone $\overline{\mathrm{Eff}}_{\mathrm{orb}}(\mathcal{X})$ is the image of the cone

[TABLE]

by the map $\lambda_{\mathrm{orb}}\colon V_{\mathrm{orb}}\to\operatorname{\mathrm{N}}_{\mathrm{orb}}^{1}(\mathcal{X})$. Moreover, it is generated by the $\#(\Sigma(1)\cup\pi_{1}^{*}(\mathcal{J}_{0}\mathcal{X}))$ elements

[TABLE]

and

[TABLE]

?proofname?.

Recall that the cone $\overline{\mathrm{Eff}}_{\mathrm{orb}}(\mathcal{X})$ is the dual cone of $\overline{\mathrm{Mov}}_{1,\mathrm{orb}}(\mathcal{X})$. If we regard $\overline{\mathrm{Mov}}_{1,\mathrm{orb}}(\mathcal{X})$ as a cone in $V_{\mathrm{orb}}$, then it is the intersection of the two cones

[TABLE]

Thus, its dual cone in $U_{\mathrm{orb}}$ is

[TABLE]

This shows that

[TABLE]

We have proved the first assertion.

To show the second assertion, we define $\Xi_{\eta}^{*}\in U_{\mathrm{orb}}$, $\eta\in\Sigma(1)\cup\pi_{0}^{*}(\mathcal{J}_{0}\mathcal{X})$ as follows:

[TABLE]

We see that $(\Xi_{\eta})_{\eta\in\Sigma(1)\cup\pi_{0}^{*}(\mathcal{J}_{0}\mathcal{X})}$ is a basis of $V_{\mathrm{orb}}$ and $(\Xi_{\eta}^{*})_{\eta\in\Sigma(1)\cup\pi_{0}^{*}(\mathcal{J}_{0}\mathcal{X})}$ is its dual basis. It follows that $\sum_{\eta}\mathbb{R}_{\geq 0}\Xi_{\eta}$ is a full-dimensional simplicial cone in $V_{\mathrm{orb}}$ and that its dual cone $\bigcap_{\eta\in\Sigma(1)\cup\pi_{0}^{*}(\mathcal{J}_{0}\mathcal{X})}\{\Xi_{\eta}\geq 0\}$ is identical to $\sum_{\eta}\mathbb{R}_{\geq 0}\Xi_{\eta}^{*}$. Since $\lambda_{\mathrm{orb}}(\mathbf{u}_{\mathcal{Y}})=u_{\mathcal{Y}}$ and $\lambda_{\mathrm{orb}}(\mathbf{u}_{\rho})=[E_{\rho}]=c_{\rho}[\mathcal{E}_{\rho}]$, we have

[TABLE]

This shows the second assertion. ∎

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