1 Introduction
The following theorem is one of the main results of this paper. Our proof of Theorem 1.1 uses the framework of the toric Mori theory developed by [Reference ReidR, Reference FujinoF1, Reference FujinoF2, Reference FujinoF4, Reference Fujino and SatoFS1], and so on.
Theorem 1.1. (Theorem 4.2.3 and Corollary 4.2.4)
Let 
$X$ be a 
$\mathbb{Q}$-Gorenstein projective toric 
$n$-fold and let 
$D$ be an ample Cartier divisor on 
$X$. Then 
$K_{X}+(n-1)D$ is pseudo-effective if and only if 
$K_{X}+(n-1)D$ is nef. In particular, if 
$X$ is Gorenstein, then
$$\begin{eqnarray}H^{0}(X,{\mathcal{O}}_{X}(K_{X}+(n-1)D))\neq 0\end{eqnarray}$$ if and only if the complete linear system 
$|K_{X}+(n-1)D|$ is basepoint-free.
This theorem was inspired by Lin’s paper (see [Reference LinLi]). Our proof of Theorem 1.1 depends on the following new estimates of lengths of extremal rays of birational type for toric varieties.
Theorem 1.2. (Theorem 3.2.1)
Let 
$f:X\rightarrow Y$ be a projective toric morphism with 
$\dim X=n$. Assume that 
$K_{X}$ is 
$\mathbb{Q}$-Cartier. Let 
$R$ be a 
$K_{X}$-negative extremal ray of 
$\operatorname{NE}(X/Y)$ and let 
$\unicode[STIX]{x1D711}_{R}:X\rightarrow W$ be the contraction morphism associated to 
$R$. We put
$$\begin{eqnarray}l(R)=\min _{[C]\in R}(-K_{X}\cdot C)\end{eqnarray}$$ and call it the length of 
$R$. Assume that 
$\unicode[STIX]{x1D711}_{R}$ is birational. Then we obtain
$$\begin{eqnarray}l(R)<d+1,\end{eqnarray}$$where
$$\begin{eqnarray}d=\max _{w\in W}\dim \unicode[STIX]{x1D711}_{R}^{-1}(w)\leqslant n-1.\end{eqnarray}$$ When 
$d=n-1$, we have a sharper inequality
$$\begin{eqnarray}l(R)\leqslant d=n-1.\end{eqnarray}$$ In particular, if 
$l(R)=n-1$, then 
$\unicode[STIX]{x1D711}_{R}:X\rightarrow W$ can be described as follows. There exists a torus invariant smooth point 
$P\in W$ such that 
$\unicode[STIX]{x1D711}_{R}:X\rightarrow W$ is a weighted blow-up at 
$P$ with the weight 
$(1,a,\ldots ,a)$ for some positive integer 
$a$. In this case, the exceptional locus 
$E$ of 
$\unicode[STIX]{x1D711}_{R}$ is a torus invariant prime divisor and is isomorphic to 
$\mathbb{P}^{n-1}$. Moreover, 
$X$ is 
$\mathbb{Q}$-factorial in a neighborhood of 
$E$.
Theorem 1.2 supplements [Reference FujinoF1, Theorem 0.1] (see also [Reference FujinoF2, Theorem 3.13]). We will see that the estimates obtained in Theorem 1.2 are the best by constructing some examples explicitly (see Examples 3.3.1 and 3.3.2). For lengths of extremal rays for nontoric varieties, see [Reference KawamataK]. As an application of Theorem 1.2, we can prove the following theorem on lengths of extremal rays for 
$\mathbb{Q}$-Gorenstein toric varieties.
Theorem 1.3. (Theorem 3.2.9)
Let 
$X$ be a 
$\mathbb{Q}$-Gorenstein projective toric 
$n$-fold with 
$\unicode[STIX]{x1D70C}(X)\geqslant 2$. Let 
$R$ be a 
$K_{X}$-negative extremal ray of 
$\operatorname{NE}(X)$ such that
$$\begin{eqnarray}l(R)=\min _{[C]\in R}(-K_{X}\cdot C)>n-1.\end{eqnarray}$$ Then the extremal contraction 
$\unicode[STIX]{x1D711}_{R}:X\rightarrow W$ associated to 
$R$ is a 
$\mathbb{P}^{n-1}$-bundle over 
$\mathbb{P}^{1}$.
As a direct easy consequence of Theorem 1.3, we obtain the following corollary, which supplements Theorem 1.1.
Corollary 1.4. (Corollary 4.2.5)
Let 
$X$ be a 
$\mathbb{Q}$-Gorenstein projective toric 
$n$-fold and let 
$D$ be an ample Cartier divisor on 
$X$. If 
$\unicode[STIX]{x1D70C}(X)\geqslant 3$, then 
$K_{X}+(n-1)D$ is always nef. More precisely, if 
$\unicode[STIX]{x1D70C}(X)\geqslant 2$ and 
$X$ is not a 
$\mathbb{P}^{n-1}$-bundle over 
$\mathbb{P}^{1}$, then 
$K_{X}+(n-1)D$ is nef.
In this paper, we also give some generalizations of Fujita’s freeness and very ampleness for toric varieties based on our powerful vanishing theorem (see [Reference FujinoF5, Reference FujinoF6]). As a very special case of our generalization of Fujita’s freeness for toric varieties (see Theorem 4.1.1), we can easily recover some parts of Lin’s theorem (see [Reference LinLi, Main theorem A]).
Theorem 1.5. (Corollary 4.1.2)
Let 
$X$ be an 
$n$-dimensional projective toric variety and let 
$D$ be an ample Cartier divisor on 
$X$. Then the reflexive sheaf 
${\mathcal{O}}_{X}(K_{X}+(n+1)D)$ is generated by its global sections.
By the same way, we can obtain a generalization of Fujita’s very ampleness for toric varieties (see Theorem 4.1.8). We note that Payne completely settled Fujita’s very ampleness conjecture for singular projective toric varieties by his clever combinatorial approach (see [Reference PayneP]). As was mentioned above, we do not use combinatorial arguments, but apply some vanishing theorems for the proof of Theorem 1.6.
Theorem 1.6. (Theorem 4.1.6)
Let 
$f:X\rightarrow Y$ be a proper surjective toric morphism, let 
$\unicode[STIX]{x1D6E5}$ be a reduced torus invariant divisor on 
$X$ such that 
$K_{X}+\unicode[STIX]{x1D6E5}$ is Cartier, and let 
$D$ be an 
$f$-ample Cartier divisor on 
$X$. Then 
${\mathcal{O}}_{X}(K_{X}+\unicode[STIX]{x1D6E5}+kD)$ is 
$f$-very ample for every 
$k\geqslant \max _{y\in Y}\dim f^{-1}(y)+2$.
For the precise statements of our generalizations of Fujita’s freeness and very ampleness for toric varieties, see Theorems 4.1.1 and 4.1.8. We omit them here since they are technically complicated.
This paper is organized as follows. In Section 2, we collect some basic definitions and results. In Section 2.1, we explain the basic concepts of the toric geometry. In Section 2.2, we recall the definitions of the Kleiman–Mori cone, the nef cone, the ample cone, and the pseudo-effective cone for toric varieties, and some related results. In Section 2.3, we explain subadjunction for 
$\mathbb{Q}$-factorial toric varieties. Section 3 is the main part of this paper. After recalling the known estimates of lengths of extremal rays for projective toric varieties in Section 3.1, we give new estimates of lengths of extremal rays of toric birational contraction morphisms in Section 3.2. In Section 3.3, we see that the estimates obtained in Section 3.2 are the best by constructing some examples explicitly. Section 4 treats Fujita’s freeness and very ampleness for toric varieties. The results in Section 4.1 depend on our powerful vanishing theorem for toric varieties and are independent of our estimates of lengths of extremal rays for toric varieties. Therefore, Section 4.1 is independent of the other parts of this paper. In Section 4.2, we discuss Lin’s problem (see [Reference LinLi]) related to Fujita’s freeness for toric varieties. We use our new estimates of lengths of extremal rays in this subsection. Section 4.3 is a supplement to Fujita’s paper: [Reference FujitaFuj]. This paper contains various supplementary results for [Reference FujitaFuj, Reference FultonFul, Reference LinLi], and so on.
We will work over an arbitrary algebraically closed field throughout this paper. For the standard notations of the minimal model program, see [Reference FujinoF7, Reference FujinoF8]. For the toric Mori theory, we recommend the reader to see [Reference ReidR], [Reference MatsukiMa, Chapter 14], [Reference FujinoF1], and [Reference Fujino and SatoFS1] (see also [Reference Cox, Little and SchenckCLS]).
2 Preliminaries
This section collects some basic definitions and results.
2.1 Basics of the toric geometry
In this subsection, we recall the basic notion of toric varieties and fix the notation. For the details, see [Reference OdaO, Reference FultonFul, Reference ReidR], or [Reference MatsukiMa, Chapter 14] (see also [Reference Cox, Little and SchenckCLS]).
2.1.1. Let 
$N\simeq \mathbb{Z}^{n}$ be a lattice of rank 
$n$. A toric variety 
$X(\unicode[STIX]{x1D6F4})$ is associated to a fan 
$\unicode[STIX]{x1D6F4}$, a correction of convex cones 
$\unicode[STIX]{x1D70E}\subset N_{\mathbb{R}}=N\otimes _{\mathbb{Z}}\mathbb{R}$ satisfying:
(i) Each convex cone
$\unicode[STIX]{x1D70E}$ is a rational polyhedral cone in the sense that there are finitely many $v_{1},\ldots ,v_{s}\in N\subset N_{\mathbb{R}}$ such that and it is strongly convex in the sense that$$\begin{eqnarray}\unicode[STIX]{x1D70E}=\{r_{1}v_{1}+\cdots +r_{s}v_{s};r_{i}\geqslant 0\}=:\langle v_{1},\ldots ,v_{s}\rangle ,\end{eqnarray}$$$$\begin{eqnarray}\unicode[STIX]{x1D70E}\cap -\unicode[STIX]{x1D70E}=\{0\}.\end{eqnarray}$$(ii) Each face
$\unicode[STIX]{x1D70F}$ of a convex cone $\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6F4}$ is again an element in $\unicode[STIX]{x1D6F4}$.(iii) The intersection of two cones in
$\unicode[STIX]{x1D6F4}$ is a face of each.
Definition 2.1.2. The dimension 
$\dim \unicode[STIX]{x1D70E}$ of a cone 
$\unicode[STIX]{x1D70E}$ is the dimension of the linear space 
$\mathbb{R}\cdot \unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}+(-\unicode[STIX]{x1D70E})$ spanned by 
$\unicode[STIX]{x1D70E}$.
We define the sublattice 
$N_{\unicode[STIX]{x1D70E}}$ of 
$N$ generated (as a subgroup) by 
$\unicode[STIX]{x1D70E}\cap N$ as follows:
$$\begin{eqnarray}N_{\unicode[STIX]{x1D70E}}:=\unicode[STIX]{x1D70E}\cap N+(-\unicode[STIX]{x1D70E}\cap N).\end{eqnarray}$$ If 
$\unicode[STIX]{x1D70E}$ is a 
$k$-dimensional simplicial cone, and 
$v_{1},\ldots ,v_{k}$ are the first lattice points along the edges of 
$\unicode[STIX]{x1D70E}$, the multiplicity of 
$\unicode[STIX]{x1D70E}$ is defined to be the index of the lattice generated by the 
$\{v_{1},\ldots ,v_{k}\}$ in the lattice 
$N_{\unicode[STIX]{x1D70E}}$;
$$\begin{eqnarray}\operatorname{mult}(\unicode[STIX]{x1D70E}):=[N_{\unicode[STIX]{x1D70E}}:\mathbb{Z}v_{1}+\cdots +\mathbb{Z}v_{k}].\end{eqnarray}$$ We note that the affine toric variety 
$X(\unicode[STIX]{x1D70E})$ associated to the cone 
$\unicode[STIX]{x1D70E}$ is smooth if and only if 
$\operatorname{mult}(\unicode[STIX]{x1D70E})=1$.
The following is a well-known fact. See, for example, [Reference MatsukiMa, Lemma 14-1-1].
Lemma 2.1.3. A toric variety 
$X(\unicode[STIX]{x1D6F4})$ is 
$\mathbb{Q}$-factorial if and only if each cone 
$\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6F4}$ is simplicial.
2.1.4. The star of a cone 
$\unicode[STIX]{x1D70F}$ can be defined abstractly as the set of cones 
$\unicode[STIX]{x1D70E}$ in 
$\unicode[STIX]{x1D6F4}$ that contain 
$\unicode[STIX]{x1D70F}$ as a face. Such cones 
$\unicode[STIX]{x1D70E}$ are determined by their images in 
$N(\unicode[STIX]{x1D70F}):=N/N_{\unicode[STIX]{x1D70F}}$, that is, by
$$\begin{eqnarray}\overline{\unicode[STIX]{x1D70E}}=\unicode[STIX]{x1D70E}+(N_{\unicode[STIX]{x1D70F}})_{\mathbb{R}}/(N_{\unicode[STIX]{x1D70F}})_{\mathbb{R}}\subset N(\unicode[STIX]{x1D70F})_{\mathbb{R}}.\end{eqnarray}$$ These cones 
$\{\overline{\unicode[STIX]{x1D70E}};\unicode[STIX]{x1D70F}\prec \unicode[STIX]{x1D70E}\}$ form a fan in 
$N(\unicode[STIX]{x1D70F})$, and we denote this fan by 
$\operatorname{Star}(\unicode[STIX]{x1D70F})$. We set 
$V(\unicode[STIX]{x1D70F})=X(\operatorname{Star}(\unicode[STIX]{x1D70F}))$, that is, the toric variety associated to the fan 
$\operatorname{Star}(\unicode[STIX]{x1D70F})$. It is well known that 
$V(\unicode[STIX]{x1D70F})$ is an 
$(n-k)$-dimensional closed toric subvariety of 
$X(\unicode[STIX]{x1D6F4})$, where 
$\dim \unicode[STIX]{x1D70F}=k$. If 
$\dim V(\unicode[STIX]{x1D70F})=1$ (resp. 
$n-1$), then we call 
$V(\unicode[STIX]{x1D70F})$ a torus invariant curve (resp. torus invariant divisor). For the details about the correspondence between 
$\unicode[STIX]{x1D70F}$ and 
$V(\unicode[STIX]{x1D70F})$, see [Reference FultonFul, 3.1 Orbits].
2.1.5. (Intersection theory for $\mathbb{Q}$-factorial toric varieties)
Assume that 
$\unicode[STIX]{x1D6F4}$ is simplicial. If 
$\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\in \unicode[STIX]{x1D6F4}$ span 
$\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6F4}$ with 
$\unicode[STIX]{x1D70E}\cap \unicode[STIX]{x1D70F}=\{0\}$, then
$$\begin{eqnarray}V(\unicode[STIX]{x1D70E})\cdot V(\unicode[STIX]{x1D70F})=\frac{\operatorname{mult}(\unicode[STIX]{x1D70E})\cdot \operatorname{mult}(\unicode[STIX]{x1D70F})}{\operatorname{mult}(\unicode[STIX]{x1D6FE})}V(\unicode[STIX]{x1D6FE})\end{eqnarray}$$ in the Chow group 
$A^{\ast }(X)_{\mathbb{Q}}$. For the details, see [Reference FultonFul, 5.1 Chow groups]. If 
$\unicode[STIX]{x1D70E}$ and 
$\unicode[STIX]{x1D70F}$ are contained in no cone of 
$\unicode[STIX]{x1D6F4}$, then 
$V(\unicode[STIX]{x1D70E})\cdot V(\unicode[STIX]{x1D70F})=0$.
2.2 Cones of divisors
In this subsection, we explain various cones of divisors and some related topics.
2.2.1. Let 
$f:X\rightarrow Y$ be a proper toric morphism; a 
$1$-cycle of 
$X/Y$ is a formal sum 
$\sum a_{i}C_{i}$ with complete curves 
$C_{i}$ in the fibers of 
$f$, and 
$a_{i}\in \mathbb{Z}$. We put
$$\begin{eqnarray}Z_{1}(X/Y):=\{1\text{-cycles of }X/Y\},\end{eqnarray}$$and
$$\begin{eqnarray}Z_{1}(X/Y)_{\mathbb{R}}:=Z_{1}(X/Y)\otimes \mathbb{R}.\end{eqnarray}$$There is a pairing
$$\begin{eqnarray}\operatorname{Pic}(X)\times Z_{1}(X/Y)_{\mathbb{R}}\rightarrow \mathbb{R}\end{eqnarray}$$ defined by 
$({\mathcal{L}},C)\mapsto \deg _{C}{\mathcal{L}}$, extended by bilinearity. We define
$$\begin{eqnarray}N^{1}(X/Y):=(\operatorname{Pic}(X)\otimes \mathbb{R})/\equiv\end{eqnarray}$$and
$$\begin{eqnarray}N_{1}(X/Y):=Z_{1}(X/Y)_{\mathbb{R}}/\equiv ,\end{eqnarray}$$ where the numerical equivalence 
$\equiv$ is by definition the smallest equivalence relation which makes 
$N^{1}$ and 
$N_{1}$ into dual spaces.
Inside 
$N_{1}(X/Y)$ there is a distinguished cone of effective 
$1$-cycles of 
$X/Y$,
$$\begin{eqnarray}\operatorname{NE}(X/Y)=\bigg\{Z\mid Z\equiv \sum a_{i}C_{i}\text{ with }a_{i}\in \mathbb{R}_{{\geqslant}0}\bigg\}\subset N_{1}(X/Y),\end{eqnarray}$$ which is usually called the Kleiman–Mori cone of 
$f:X\rightarrow Y$. It is known that 
$\operatorname{NE}(X/Y)$ is a rational polyhedral cone. A face 
$F\prec \operatorname{NE}(X/Y)$ is called an extremal face in this case. A one-dimensional extremal face is called an extremal ray.
We define the relative Picard number 
$\unicode[STIX]{x1D70C}(X/Y)$ by
$$\begin{eqnarray}\unicode[STIX]{x1D70C}(X/Y):=\dim _{\mathbb{Q}}N^{1}(X/Y)<\infty .\end{eqnarray}$$ An element 
$D\in N^{1}(X/Y)$ is called 
$f$-nef if 
$D\geqslant 0$ on 
$\operatorname{NE}(X/Y)$.
If 
$X$ is complete and 
$Y$ is a point, then we write 
$\operatorname{NE}(X)$ and 
$\unicode[STIX]{x1D70C}(X)$ for 
$\operatorname{NE}(X/Y)$ and 
$\unicode[STIX]{x1D70C}(X/Y)$, respectively. We note that 
$N_{1}(X/Y)\subset N_{1}(X)$, and 
$N^{1}(X/Y)$ is the corresponding quotient of 
$N^{1}(X)$.
From now on, we assume that 
$X$ is complete. We define the nef cone 
$\operatorname{Nef}(X)$, the ample cone 
$\operatorname{Amp}(X)$, and the pseudo-effective cone 
$\operatorname{PE}(X)$ in 
$N^{1}(X)$ as follows.
$$\begin{eqnarray}\displaystyle & \displaystyle \operatorname{Nef}(X)=\{D\mid D\text{ is nef}\}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \operatorname{Amp}(X)=\{D\mid D\text{ is ample}\} & \displaystyle \nonumber\end{eqnarray}$$and
$$\begin{eqnarray}\operatorname{PE}(X)=\left\{D\biggm\vert\begin{array}{@{}l@{}}D\equiv \sum a_{i}D_{i}\text{ such that }D_{i}\text{ is an effective}\\ \text{Cartier divisor and }a_{i}\in \mathbb{R}_{{\geqslant}0}\text{ for every }i\end{array}\right\}.\end{eqnarray}$$It is easy to see that
$$\begin{eqnarray}\operatorname{Amp}(X)\subset \operatorname{Nef}(X)\subset \operatorname{PE}(X).\end{eqnarray}$$The reader can find various examples of cones of divisors and curves in [Reference FujinoF3, Reference Fujino and PayneFP, Reference Fujino and SatoFS2].
Lemma 2.2.2. Let 
$X$ be a complete toric variety and let 
$D$ be a 
$\mathbb{Q}$-Cartier 
$\mathbb{Q}$-divisor on 
$X$. Then 
$D$ is pseudo-effective if and only if 
$\unicode[STIX]{x1D705}(X,D)\geqslant 0$, that is, there exists a positive integer 
$m$ such that 
$mD$ is Cartier and that
$$\begin{eqnarray}H^{0}(X,{\mathcal{O}}_{X}(mD))\neq 0.\end{eqnarray}$$ More generally, 
$g^{\ast }D$ is pseudo-effective for some projective birational toric morphism 
$g:Z\rightarrow X$ from a smooth projective toric variety 
$Z$ if and only if 
$\unicode[STIX]{x1D705}(X,D)\geqslant 0$.
Proof. It is sufficient to prove that 
$\unicode[STIX]{x1D705}(X,D)\geqslant 0$ when 
$g^{\ast }D$ is pseudo-effective. By replacing 
$X$ and 
$D$ with 
$Z$ and 
$g^{\ast }D$, we may assume that 
$X$ is a smooth projective toric variety. In this case, it is easy to see that 
$\operatorname{PE}(X)$ is spanned by the numerical equivalence classes of torus invariant prime divisors (see, for example, [Reference Cox, Little and SchenckCLS, Lemma 15.1.8]). Therefore, we can write 
$D\equiv \sum _{i}a_{i}D_{i}$ where 
$D_{i}$ is a torus invariant prime divisor and 
$a_{i}\in \mathbb{Q}_{{>}0}$ for every 
$i$ since 
$D$ is a 
$\mathbb{Q}$-divisor. Thus, we obtain 
$D{\sim}_{\mathbb{Q}}\sum _{i}a_{i}D_{i}\geqslant 0$. This implies 
$\unicode[STIX]{x1D705}(X,D)\geqslant 0$.◻
2.2.3. Let 
$X$ be a complete toric variety and let 
$g:Z\rightarrow X$ be a projective birational toric morphism from a smooth projective toric variety 
$Z$. Then
$$\begin{eqnarray}g^{\ast }:\operatorname{Pic}(X)\rightarrow \operatorname{Pic}(Z)\end{eqnarray}$$induces a natural inclusion
$$\begin{eqnarray}N^{1}(X){\hookrightarrow}N^{1}(Z).\end{eqnarray}$$ By this inclusion, we can see 
$N^{1}(X)$ as a linear subspace of 
$N^{1}(Z)$. It is well known that 
$\operatorname{PE}(Z)$ is a rational polyhedral cone in 
$N^{1}(Z)$ (see, for example, [Reference Cox, Little and SchenckCLS, Lemma 15.1.8]). Note that the inclusion 
$\operatorname{PE}(X)\subset N^{1}(X)\cap \operatorname{PE}(Z)$ is obvious. The opposite inclusion 
$\operatorname{PE}(X)\supset N^{1}(X)\cap \operatorname{PE}(Z)$ follows from Lemma 2.2.2. Anyway, the equality
$$\begin{eqnarray}\operatorname{PE}(X)=N^{1}(X)\cap \operatorname{PE}(Z)\end{eqnarray}$$holds. In particular, we have the following statement.
Proposition 2.2.4. Let 
$X$ be a complete toric variety. Then 
$\operatorname{PE}(X)$ is a rational polyhedral cone in 
$N^{1}(X)$.
The following lemma is well known and is very important. We will use it in the subsequent sections repeatedly.
Lemma 2.2.5. Let 
$f:X\rightarrow Y$ be a proper toric morphism and let 
$D$ be an 
$f$-nef Cartier divisor on 
$X$. Then 
$D$ is 
$f$-free, that is,
$$\begin{eqnarray}f^{\ast }f_{\ast }{\mathcal{O}}_{X}(D)\rightarrow {\mathcal{O}}_{X}(D)\end{eqnarray}$$is surjective.
Proof. See, for example, [Reference NakayamaN, Chapter VI. 1.13. Lemma]. ◻
We close this subsection with an easy example. It is well known that 
$\operatorname{NE}(X)$ is spanned by the numerical equivalence classes of torus invariant irreducible curves. However, the dual cone 
$\operatorname{Nef}(X)$ of 
$\operatorname{NE}(X)$ is not always spanned by the numerical equivalence classes of torus invariant prime divisors.
Example 2.2.6. We consider 
$\mathbb{P}^{1}\times \mathbb{P}^{1}$. Let 
$p_{i}:\mathbb{P}^{1}\times \mathbb{P}^{1}\rightarrow \mathbb{P}^{1}$ be the 
$i$th projection for 
$i=1,2$. Let 
$D_{1},D_{2}$ (resp. 
$D_{3},D_{4}$) be the torus invariant curves in the fibers of 
$p_{1}$ (resp. 
$p_{2}$). Let 
$X\rightarrow \mathbb{P}^{1}\times \mathbb{P}^{1}$ be the blow-up at the point 
$P=D_{1}\cap D_{3}$ and let 
$E$ be the exceptional curve on 
$X$. Let 
$D_{i}^{\prime }$ denote the strict transform of 
$D_{i}$ on 
$X$ for all 
$i$. Then 
$\operatorname{NE}(X)$ is spanned by the numerical equivalence classes of 
$E,D_{1}^{\prime }$, and 
$D_{3}^{\prime }$. On the other hand, 
$\operatorname{Nef}(X)\subset N^{1}(X)$ is spanned by 
$D_{2}^{\prime },D_{4}^{\prime }$, and 
$D_{1}^{\prime }+D_{3}^{\prime }+E$. Therefore, the extremal ray of 
$\operatorname{Nef}(X)$ is not necessarily spanned by a torus invariant prime divisor.
2.3 Subadjunction
In this subsection, we quickly explain subadjunction for 
$\mathbb{Q}$-factorial toric varieties for the reader’s convenience. We note that subadjunction plays an important role in the theory of minimal models (see, for example, [Reference FujinoF7, Section 14. Shokurov’s differents]).
Lemma 2.3.1. (Subadjunction)
Let 
$X$ be a 
$\mathbb{Q}$-factorial toric variety and let 
$\{{D_{i}\}}_{i\in I}$ be the set of all torus invariant prime divisors on 
$X$. We consider 
$D=\sum _{i\in I}d_{i}D_{i}$, where 
$d_{i}\in \mathbb{Q}$ and 
$0\leqslant d_{i}\leqslant 1$ for every 
$i$. Since 
$X$ is a toric variety, we can put 
$K_{X}=-\!\sum _{i\in I}D_{i}$. We assume 
$d_{i_{0}}=1$ for some 
$i_{0}\in I$. We put 
$S=D_{i_{0}}$. Let 
$\{{B_{j}\}}_{j\in J}$ be the set of all torus invariant prime divisors on 
$S$. Then the following formula
$$\begin{eqnarray}(K_{X}+D)|_{S}=K_{S}+\mathop{\sum }_{j\in J}b_{j}B_{i}\end{eqnarray}$$ holds, where 
$K_{S}=-\!\sum _{j\in J}B_{j}$, 
$b_{j}\in \mathbb{Q}$ and 
$0\leqslant b_{j}\leqslant 1$ for every 
$j$. Moreover, 
$b_{j}=1$ holds in (2.1) if and only if there exists 
$i(j)\in I$ such that 
$d_{i(j)}=1$ and that 
$B_{j}=D_{i(j)}\cap S$. We note that 
$\sum _{j\in J}b_{j}B_{j}$ in (2.1) is usually called a different.
Proof. We note that
$$\begin{eqnarray}K_{X}+D=K_{X}+\mathop{\sum }_{i\in I}D_{i}-\mathop{\sum }_{i\in I}(1-d_{i})D_{i}=-\!\mathop{\sum }_{i\in I}(1-d_{i})D_{i}.\end{eqnarray}$$Therefore, we have
$$\begin{eqnarray}(K_{X}+D)|_{S}=-\!\mathop{\sum }_{i\in I}(1-d_{i})D_{i}\cdot S=K_{S}+\mathop{\sum }_{j\in J}B_{j}-\mathop{\sum }_{i\in I}(1-d_{i})D_{i}\cdot S.\end{eqnarray}$$We put
$$\begin{eqnarray}\mathop{\sum }_{j\in J}b_{j}B_{j}=\mathop{\sum }_{j\in J}B_{j}-\mathop{\sum }_{i\in I}(1-d_{i})D_{i}\cdot S.\end{eqnarray}$$ Then we obtain 
$b_{j}\in \mathbb{Q}$ and 
$0\leqslant b_{j}\leqslant 1$ for every 
$j$ by 2.1.5. By (2.2), it is easy to see that 
$b_{j}=1$ holds if and only if there exists 
$i(j)\in I$ such that 
$d_{i(j)}=1$ and 
$D_{i(j)}\cap S=B_{j}$.◻
3 Lengths of extremal rays
In this section, we discuss some estimates of lengths of extremal rays of projective toric morphisms.
3.1 Quick review of the known estimates
In this subsection, we recall the known estimates of lengths of extremal rays for toric varieties. The first result is [Reference FujinoF1, Theorem 0.1] (see also [Reference FujinoF2, Theorem 3.13]).
Theorem 3.1.1. Let 
$f:X\rightarrow Y$ be a projective toric morphism with 
$\dim X=n$ and let 
$\unicode[STIX]{x1D6E5}=\sum \unicode[STIX]{x1D6FF}_{i}\unicode[STIX]{x1D6E5}_{i}$ be an 
$\mathbb{R}$-divisor on 
$X$ such that 
$\unicode[STIX]{x1D6E5}_{i}$ is a torus invariant prime divisor and 
$0\leqslant \unicode[STIX]{x1D6FF}_{i}\leqslant 1$ for every 
$i$. Assume that 
$K_{X}+\unicode[STIX]{x1D6E5}$ is 
$\mathbb{R}$-Cartier. Let 
$R$ be an extremal ray of 
$\operatorname{NE}(X/Y)$. Then there exists a curve 
$C$ on 
$X$ such that 
$[C]\in R$ and
$$\begin{eqnarray}-(K_{X}+\unicode[STIX]{x1D6E5})\cdot C\leqslant n+1.\end{eqnarray}$$ More precisely, we can choose 
$C$ such that
$$\begin{eqnarray}-(K_{X}+\unicode[STIX]{x1D6E5})\cdot C\leqslant n\end{eqnarray}$$ unless 
$X\simeq \mathbb{P}^{n}$ and 
$\sum \unicode[STIX]{x1D6FF}_{i}<1$. We note that if 
$X$ is complete then we can make 
$C$ a torus invariant curve on 
$X$.
Our proof of Theorems 3.1.1 and 3.2.1 heavily depends on Reid’s description of toric extremal contraction morphisms (see [Reference ReidR] and [Reference MatsukiMa, Chapter 14]).
3.1.2. (Reid’s description of toric extremal contraction morphisms)
Let 
$f:X\rightarrow Y$ be a projective surjective toric morphism from a complete 
$\mathbb{Q}$-factorial toric 
$n$-fold and let 
$R$ be an extremal ray of 
$\operatorname{NE}(X/Y)$. Let 
$\unicode[STIX]{x1D711}_{R}:X\rightarrow W$ be the extremal contraction associated to 
$R$. We write
$$\begin{eqnarray}\begin{array}{@{}cccc@{}} & A & \longrightarrow & B\\ & \cap & & \cap \\ \unicode[STIX]{x1D711}_{R}: & X & \longrightarrow & W,\\ \end{array}\end{eqnarray}$$ where 
$A$ is the exceptional locus of 
$\unicode[STIX]{x1D711}_{R}$ and 
$B$ is the image of 
$A$ by 
$\unicode[STIX]{x1D711}_{R}$. Then there exist torus invariant prime divisors 
$E_{1},\ldots ,E_{\unicode[STIX]{x1D6FC}}$ on 
$X$ with 
$0\leqslant \unicode[STIX]{x1D6FC}\leqslant n-1$ such that 
$E_{i}$ is negative on 
$R$ for 
$1\leqslant i\leqslant \unicode[STIX]{x1D6FC}$ and that 
$A$ is 
$E_{1}\cap \cdots \cap E_{\unicode[STIX]{x1D6FC}}$. In particular, 
$A$ is an irreducible torus invariant subvariety of 
$X$ with 
$\dim A=n-\unicode[STIX]{x1D6FC}$. Note that 
$\unicode[STIX]{x1D6FC}=0$ if and only if 
$A=X$, that is, 
$\unicode[STIX]{x1D711}_{R}$ is a Fano contraction. There are torus invariant prime divisors 
$E_{\unicode[STIX]{x1D6FD}+1},\ldots ,E_{n+1}$ on 
$X$ with 
$\unicode[STIX]{x1D6FC}\leqslant \unicode[STIX]{x1D6FD}\leqslant n-1$ such that 
$E_{i}$ is positive on 
$R$ for 
$\unicode[STIX]{x1D6FD}+1\leqslant i\leqslant n+1$. Let 
$F$ be a general fiber of 
$A\rightarrow B$. Then 
$F$ is a 
$\mathbb{Q}$-factorial toric Fano variety with 
$\unicode[STIX]{x1D70C}(F)=1$ and 
$\dim F=n-\unicode[STIX]{x1D6FD}$. The divisors 
$E_{\unicode[STIX]{x1D6FD}+1}|_{F},\ldots ,E_{n+1}|_{F}$ define all the torus invariant prime divisors on 
$F$. In particular, 
$B$ is an irreducible torus invariant subvariety of 
$W$ with 
$\dim B=\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6FC}$. When 
$X$ is not complete, we can reduce it to the case where 
$X$ is complete by the equivariant completion theorem in [Reference FujinoF2]. For the details, see [Reference SatoS].
3.1.3. We quickly review the idea of the proof of Theorem 3.1.1 in [Reference FujinoF1]. We will use the same idea in the proof of Theorem 3.2.1. By replacing 
$X$ with its projective 
$\mathbb{Q}$-factorialization, we may assume that 
$X$ is 
$\mathbb{Q}$-factorial. Let 
$R$ be an extremal ray of 
$\operatorname{NE}(X/Y)$. Then we consider the extremal contraction 
$\unicode[STIX]{x1D711}_{R}:X\rightarrow W$ associated to 
$R$. If 
$X$ is not projective, then we can reduce it to the case where 
$X$ is projective by the equivariant completion theorem (see [Reference FujinoF2]). By Reid’s combinatorial description of 
$\unicode[STIX]{x1D711}_{R}$, any fiber 
$F$ of 
$\unicode[STIX]{x1D711}_{R}$ is a 
$\mathbb{Q}$-factorial projective toric variety with 
$\unicode[STIX]{x1D70C}(F)=1$. By subadjunction (see Lemma 2.3.1), we can compare 
$-(K_{X}+\unicode[STIX]{x1D6E5})\cdot C$ with 
$-K_{F}\cdot C$, where 
$C$ is a curve on 
$F$. So, the key ingredient of the proof of Theorem 3.1.1 is the following proposition.
Proposition 3.1.4. Let 
$X$ be a 
$\mathbb{Q}$-factorial projective toric 
$n$-fold with 
$\unicode[STIX]{x1D70C}(X)=1$. Assume that 
$-K_{X}\cdot C>n$ for every integral curve 
$C$ on 
$X$. Then 
$X\simeq \mathbb{P}^{n}$.
For the proof, see [Reference FujinoF1, Proposition 2.9]. Our proof heavily depends on the calculation described in 3.1.8.
3.1.5. (Supplements to [Reference FujinoF4])
By the same arguments as in the proof of Proposition 3.1.4, we can obtain the next proposition, which is nothing but [Reference FujinoF4, Proposition 2.1].
Proposition 3.1.6. Let 
$X$ be a 
$\mathbb{Q}$-factorial projective toric 
$n$-fold with 
$\unicode[STIX]{x1D70C}(X)=1$ such that 
$X\not \simeq \mathbb{P}^{n}$. Assume that 
$-K_{X}\cdot C\geqslant n$ for every integral curve 
$C$ on 
$X$. Then 
$X$ is isomorphic to the weighted projective space 
$\mathbb{P}(1,1,2,\ldots ,2)$.
The following proposition, which is missing in [Reference FujinoF4], is a characterization of hyperquadrics for toric varieties (see Corollary of [Reference Kobayashi and OchiaiKO, Theorem 2.1]). This proposition says that the results in [Reference FujinoF4] are compatible with [Reference FujitaFuj, Theorem 2(a)].
Proposition 3.1.7. Let 
$X$ be a projective toric 
$n$-fold with 
$n\geqslant 2$. We assume that 
$-K_{X}\equiv nD$ for some Cartier divisor 
$D$ on 
$X$ and 
$\unicode[STIX]{x1D70C}(X)=1$. Then 
$D$ is very ample and 
$\unicode[STIX]{x1D6F7}_{|D|}:X{\hookrightarrow}\mathbb{P}^{n+1}$ embeds 
$X$ into 
$\mathbb{P}^{n+1}$ as a hyperquadric.
Proof. By [Reference FujinoF4, Theorem 3.2, Remark 3.3, and Theorem 3.4], there exists a crepant toric resolution 
$\unicode[STIX]{x1D711}:Y\rightarrow X$, where 
$Y=\mathbb{P}_{\mathbb{P}^{1}}({\mathcal{O}}_{\mathbb{P}^{1}}\,\oplus \,\cdots \,\oplus \,{\mathcal{O}}_{\mathbb{P}^{1}}\,\oplus \,{\mathcal{O}}_{\mathbb{P}^{1}}(2))$ or 
$Y=\mathbb{P}_{\mathbb{P}^{1}}({\mathcal{O}}_{\mathbb{P}^{1}}\,\oplus \,\cdots \,\oplus \,{\mathcal{O}}_{\mathbb{P}^{1}}\,\oplus \,{\mathcal{O}}_{\mathbb{P}^{1}}(1)\,\oplus \,{\mathcal{O}}_{\mathbb{P}^{1}}(1))$. We note that 
$X=\mathbb{P}(1,1,2,\ldots ,2)$ when 
$Y=\mathbb{P}_{\mathbb{P}^{1}}({\mathcal{O}}_{\mathbb{P}^{1}}\,\oplus \,\cdots \,\oplus \,{\mathcal{O}}_{\mathbb{P}^{1}}\,\oplus \,{\mathcal{O}}_{\mathbb{P}^{1}}(2))$. We also note that 
$X$ is not 
$\mathbb{Q}$-factorial if 
$Y=\mathbb{P}_{\mathbb{P}^{1}}({\mathcal{O}}_{\mathbb{P}^{1}}\,\oplus \,\cdots \,\oplus \,{\mathcal{O}}_{\mathbb{P}^{1}}\,\oplus \,{\mathcal{O}}_{\mathbb{P}^{1}}(1)\,\oplus \,{\mathcal{O}}_{\mathbb{P}^{1}}(1))$. Let 
${\mathcal{O}}_{Y}(1)$ be the tautological line bundle of the 
$\mathbb{P}^{n-1}$-bundle 
$Y$ over 
$\mathbb{P}^{1}$. Then we have 
${\mathcal{O}}_{Y}(-K_{Y})\simeq {\mathcal{O}}_{Y}(n)$. We can directly check that 
$\dim H^{0}(Y,{\mathcal{O}}_{Y}(1))=n+2$. We consider 
$\unicode[STIX]{x1D6F7}_{|{\mathcal{O}}_{Y}(1)|}:Y\rightarrow \mathbb{P}^{n+1}$. By construction,
$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{|{\mathcal{O}}_{Y}(1)|}:Y\overset{\unicode[STIX]{x1D711}}{\longrightarrow }X\overset{\unicode[STIX]{x1D70B}}{\longrightarrow }\mathbb{P}^{n+1}\end{eqnarray}$$ is the Stein factorization of 
$\unicode[STIX]{x1D6F7}_{|{\mathcal{O}}_{Y}(1)|}:Y\rightarrow \mathbb{P}^{n+1}$. By construction again, we have 
${\mathcal{O}}_{Y}(1)\simeq \unicode[STIX]{x1D711}^{\ast }{\mathcal{O}}_{X}(D)$. Since we can directly check that
$$\begin{eqnarray}\text{Sym}^{k}H^{0}(Y,{\mathcal{O}}_{Y}(1))\rightarrow H^{0}(Y,{\mathcal{O}}_{Y}(k))\end{eqnarray}$$ is surjective for every 
$k\in \mathbb{Z}_{{>}0}$, we see that
$$\begin{eqnarray}\text{Sym}^{k}H^{0}(X,{\mathcal{O}}_{X}(D))\rightarrow H^{0}(X,{\mathcal{O}}_{X}(kD))\end{eqnarray}$$ is also surjective for every 
$k\in \mathbb{Z}_{{>}0}$. This means that 
${\mathcal{O}}_{X}(D)$ is very ample. In particular, 
$\unicode[STIX]{x1D70B}:X\rightarrow \mathbb{P}^{n+1}$ is nothing but the embedding 
$\unicode[STIX]{x1D6F7}_{|D|}:X{\hookrightarrow}\mathbb{P}^{n+1}$. Since 
$D^{n}=({\mathcal{O}}_{Y}(1))^{n}=2$, 
$X$ is a hyperquadric in 
$\mathbb{P}^{n+1}$.◻
As was mentioned above, the following calculation plays an important role in the proof of Proposition 3.1.4.
3.1.8. (Fake weighted projective spaces)
Now we fix 
$N\simeq \mathbb{Z}^{n}$. Let 
$\{v_{1},\ldots ,v_{n+1}\}$ be a set of primitive vectors of 
$N$ such that 
$N_{\mathbb{R}}=\sum _{i}\mathbb{R}_{{\geqslant}0}v_{i}$. We define 
$n$-dimensional cones
$$\begin{eqnarray}\unicode[STIX]{x1D70E}_{i}:=\langle v_{1},\ldots ,v_{i-1},v_{i+1},\ldots ,v_{n+1}\rangle\end{eqnarray}$$ for 
$1\leqslant i\leqslant n+1$. Let 
$\unicode[STIX]{x1D6F4}$ be the complete fan generated by 
$n$-dimensional cones 
$\unicode[STIX]{x1D70E}_{i}$ and their faces for all 
$i$. Then we obtain a complete toric variety 
$X=X(\unicode[STIX]{x1D6F4})$ with Picard number 
$\unicode[STIX]{x1D70C}(X)=1$. We call it a 
$\mathbb{Q}$-factorial toric Fano variety with Picard number one. It is sometimes called a fake weighted projective space. We define 
$(n-1)$-dimensional cones 
$\unicode[STIX]{x1D707}_{i,j}=\unicode[STIX]{x1D70E}_{i}\cap \unicode[STIX]{x1D70E}_{j}$ for 
$i\neq j$. We can write 
$\sum _{i}a_{i}v_{i}=0$, where 
$a_{i}\in \mathbb{Z}_{{>}0}$, 
$\gcd (a_{1},\ldots ,a_{n+1})=1$, and 
$a_{1}\leqslant a_{2}\leqslant \cdots \leqslant a_{n+1}$ by changing the order. Then we obtain
$$\begin{eqnarray}\displaystyle & \displaystyle 0<V(v_{n+1})\cdot V(\unicode[STIX]{x1D707}_{n,n+1})=\frac{\operatorname{mult}(\unicode[STIX]{x1D707}_{n,n+1})}{\operatorname{mult}(\unicode[STIX]{x1D70E}_{n})}\leqslant 1, & \displaystyle \nonumber\\ \displaystyle & \displaystyle V(v_{i})\cdot V(\unicode[STIX]{x1D707}_{n,n+1})=\frac{a_{i}}{a_{n+1}}\cdot \frac{\operatorname{mult}(\unicode[STIX]{x1D707}_{n,n+1})}{\operatorname{mult}(\unicode[STIX]{x1D70E}_{n})}, & \displaystyle \nonumber\end{eqnarray}$$and
$$\begin{eqnarray}\displaystyle -K_{X}\cdot V(\unicode[STIX]{x1D707}_{n,n+1}) & = & \displaystyle \mathop{\sum }_{i=1}^{n+1}V(v_{i})\cdot V(\unicode[STIX]{x1D707}_{n,n+1})\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{a_{n+1}}\biggl(\mathop{\sum }_{i=1}^{n+1}a_{i}\biggr)\frac{\operatorname{mult}(\unicode[STIX]{x1D707}_{n,n+1})}{\operatorname{mult}(\unicode[STIX]{x1D70E}_{n})}\leqslant n+1.\nonumber\end{eqnarray}$$We note that
$$\begin{eqnarray}\frac{\operatorname{mult}(\unicode[STIX]{x1D70E}_{n})}{\operatorname{mult}(\unicode[STIX]{x1D707}_{n,n+1})}\in \mathbb{Z}_{{>}0}.\end{eqnarray}$$ For the procedure to compute intersection numbers, see 2.1.5 or [Reference FultonFul, page 100]. If 
$-K_{X}\cdot V(\unicode[STIX]{x1D707}_{n,n+1})=n+1$, then 
$a_{i}=1$ for every 
$i$ and 
$\operatorname{mult}(\unicode[STIX]{x1D707}_{n,n+1})=\operatorname{mult}(\unicode[STIX]{x1D70E}_{n})$.
We note that the above calculation plays crucial roles in [Reference FujinoF1, Reference FujinoF4, Reference Fujino and IshitsukaFI], and this paper (see the proof of Theorem 3.2.1, and so on).
Lemma 3.1.9. We use the same notation as in 3.1.8. We consider the sublattice 
$N^{\prime }$ of 
$N$ spanned by 
$\{v_{1},\ldots ,v_{n+1}\}$. Then the natural inclusion 
$N^{\prime }\rightarrow N$ induces a finite toric morphism 
$f:X^{\prime }\rightarrow X$ from a weighted projective space 
$X^{\prime }$ such that 
$f$ is étale in codimension one. In particular, 
$X(\unicode[STIX]{x1D6F4})$ is a weighted projective space if and only if 
$\{v_{1},\ldots ,v_{n+1}\}$ generates 
$N$.
We note the above elementary lemma. Example 3.1.10 shows that there are many fake weighted projective spaces which are not weighted projective spaces.
Example 3.1.10. We put 
$N=\mathbb{Z}^{3}$. Let 
$\{e_{1},e_{2},e_{3}\}$ be the standard basis of 
$N$. We put 
$v_{1}=e_{1}$, 
$v_{2}=e_{2}$, 
$v_{3}=e_{3}$, and 
$v_{4}=-e_{1}-e_{2}-e_{3}$. The fan 
$\unicode[STIX]{x1D6F4}$ is the subdivision of 
$N_{\mathbb{R}}$ by 
$\{v_{1},v_{2},v_{3},v_{4}\}$. Then 
$Y=X(\unicode[STIX]{x1D6F4})\simeq \mathbb{P}^{3}$. We consider a new lattice
$$\begin{eqnarray}N^{\dagger }=N+({\textstyle \frac{1}{2}},{\textstyle \frac{1}{2}},0)\mathbb{Z}.\end{eqnarray}$$ The natural inclusion 
$N\rightarrow N^{\dagger }$ induces a finite toric morphism 
$Y\rightarrow X$, which is étale in codimension one. It is easy to see that 
$K_{X}$ is Cartier and 
$-K_{X}\sim 4D_{4}$, where 
$D_{4}=V(v_{4})$ is not Cartier but 
$2D_{4}$ is Cartier. Since 
$\{v_{1},v_{2},v_{3},v_{4}\}$ does not span the lattice 
$N^{\dagger }$, 
$X$ is not a weighted projective space. Of course, 
$X$ is a fake weighted projective space.
3.2 New estimate of lengths of extremal rays
The following theorem is one of the main theorems of this paper, in which we prove new estimates of 
$K_{X}$-negative extremal rays of birational type. We will see that they are the best by Examples 3.3.1 and 3.3.2.
Theorem 3.2.1. (Lengths of extremal rays of birational type, Theorem 1.2)
Let 
$f:X\rightarrow Y$ be a projective toric morphism with 
$\dim X=n$. Assume that 
$K_{X}$ is 
$\mathbb{Q}$-Cartier. Let 
$R$ be a 
$K_{X}$-negative extremal ray of 
$\operatorname{NE}(X/Y)$ and let 
$\unicode[STIX]{x1D711}_{R}:X\rightarrow W$ be the contraction morphism associated to 
$R$. We put
$$\begin{eqnarray}l(R)=\min _{[C]\in R}(-K_{X}\cdot C)\end{eqnarray}$$ and call it the length of 
$R$. Assume that 
$\unicode[STIX]{x1D711}_{R}$ is birational. Then we obtain
$$\begin{eqnarray}l(R)<d+1,\end{eqnarray}$$where
$$\begin{eqnarray}d=\max _{w\in W}\dim \unicode[STIX]{x1D711}_{R}^{-1}(w)\leqslant n-1.\end{eqnarray}$$ When 
$d=n-1$, we have a sharper inequality
$$\begin{eqnarray}l(R)\leqslant d=n-1.\end{eqnarray}$$ In particular, if 
$l(R)=n-1$, then 
$\unicode[STIX]{x1D711}_{R}:X\rightarrow W$ can be described as follows. There exists a torus invariant smooth point 
$P\in W$ such that 
$\unicode[STIX]{x1D711}_{R}:X\rightarrow W$ is a weighted blow-up at 
$P$ with the weight 
$(1,a,\ldots ,a)$ for some positive integer 
$a$. In this case, the exceptional locus 
$E$ of 
$\unicode[STIX]{x1D711}_{R}$ is a torus invariant prime divisor and is isomorphic to 
$\mathbb{P}^{n-1}$. Moreover, 
$X$ is 
$\mathbb{Q}$-factorial in a neighborhood of 
$E$.
The idea of the proof of Theorem 3.2.1 is the same as that of Theorem 3.1.1.
Proof of Theorem 3.2.1.
In Step 1, we will explain how to reduce problems to the case where 
$X$ is 
$\mathbb{Q}$-factorial. Then we will prove the inequality 
$l(R)<d+1$ in Step 2. In Step 3, we will treat the case where 
$X$ is 
$\mathbb{Q}$-factorial and 
$l(R)\geqslant n-1$. Finally, in Step 4, we will treat the case where 
$l(R)\geqslant n-1$ under the assumption that 
$X$ is not necessarily 
$\mathbb{Q}$-factorial.
Step 1. In this step, we will explain how to reduce problems to the case where 
$X$ is 
$\mathbb{Q}$-factorial.
Without loss of generality, we may assume that 
$W=Y$. Let 
$\unicode[STIX]{x1D70B}:\widetilde{X}\rightarrow X$ be a small projective 
$\mathbb{Q}$-factorialization (see, for example, [Reference FujinoF1, Corollary 5.9]). Then we can take an extremal ray 
$\widetilde{R}$ of 
$\operatorname{NE}(\widetilde{X}/W)$ and construct the following commutative diagram
where 
$\unicode[STIX]{x1D711}_{\widetilde{R}}$ is the contraction morphism associated to 
$\widetilde{R}$. We note that 
$\unicode[STIX]{x1D711}_{\widetilde{R}}$ must be small when 
$\unicode[STIX]{x1D711}_{R}$ is small, because the composition of small morphisms 
$\unicode[STIX]{x1D70B}$ and 
$\unicode[STIX]{x1D711}_{R}$ is also a small morphism. We write
where 
$\widetilde{A}$ is the exceptional locus of 
$\unicode[STIX]{x1D711}_{\widetilde{R}}$ and 
$\widetilde{B}$ is the image of 
$\widetilde{A}$. Let 
$\widetilde{F}$ be a general fiber of 
$\widetilde{A}\rightarrow \widetilde{B}$. Then 
$\widetilde{F}$ is a fake weighted projective space as in 3.1.2, that is, 
$\widetilde{F}$ is a 
$\mathbb{Q}$-factorial toric Fano variety with Picard number one. Since 
$\unicode[STIX]{x1D70C}(\widetilde{F})=1$, 
$\unicode[STIX]{x1D70B}:\widetilde{F}\rightarrow F:=\unicode[STIX]{x1D70B}(\widetilde{F})$ is finite. Therefore, by definition, 
$\dim \widetilde{F}=\dim F\leqslant d$ since 
$\unicode[STIX]{x1D711}_{R}(F)$ is a point. Let 
$\widetilde{C}$ be a curve in 
$\widetilde{F}$ and let 
$C$ be the image of 
$\widetilde{C}$ by 
$\unicode[STIX]{x1D70B}$ with the reduced scheme structure. Then we obtain
$$\begin{eqnarray}-K_{\widetilde{X}}\cdot \widetilde{C}=-\unicode[STIX]{x1D70B}^{\ast }K_{X}\cdot \widetilde{C}=-mK_{X}\cdot C,\end{eqnarray}$$ where 
$m$ is the mapping degree of 
$\widetilde{C}\rightarrow C$. Thus, if 
$-K_{\widetilde{X}}\cdot \widetilde{C}$ satisfies the desired inequality, then 
$-K_{X}\cdot C$ also satisfies the same inequality. Therefore, for the proof of 
$l(R)<d+1$, we may assume that 
$X$ is 
$\mathbb{Q}$-factorial and 
$W=Y$ by replacing 
$X$ and 
$Y$ with 
$\widetilde{X}$ and 
$\widetilde{W}$, respectively.
Step 2. In this step, we will prove the desired inequality 
$l(R)<d+1$ under the assumption that 
$X$ is 
$\mathbb{Q}$-factorial.
We write
$$\begin{eqnarray}\begin{array}{@{}cccc@{}} & A & \longrightarrow & B\\ & \cap & & \cap \\ \unicode[STIX]{x1D711}_{R}: & X & \longrightarrow & W,\\ \end{array}\end{eqnarray}$$ where 
$A$ is the exceptional locus of 
$\unicode[STIX]{x1D711}_{R}$ and 
$B$ is the image of 
$A$. We note that 
$A$ is irreducible (see 3.1.2). We put 
$\dim A=n-\unicode[STIX]{x1D6FC}$ and 
$\dim B=\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6FC}$ as in 3.1.2. Let 
$F$ be a general fiber of 
$A\rightarrow B$. Then we know that 
$F$ is a 
$\mathbb{Q}$-factorial toric Fano variety with Picard number one and that there exist torus invariant prime divisors 
$E_{1},\ldots ,E_{\unicode[STIX]{x1D6FC}}$ on 
$X$ such that 
$E_{i}$ is negative on 
$R$ for every 
$i$ and 
$A$ is 
$E_{1}\cap \cdots \cap E_{\unicode[STIX]{x1D6FC}}$ (see 3.1.2). By using subadjunction (see Lemma 2.3.1) repeatedly, we have
$$\begin{eqnarray}(K_{X}+E_{1}+\cdots +E_{\unicode[STIX]{x1D6FC}})|_{A}=K_{A}+D\end{eqnarray}$$ for some effective 
$\mathbb{Q}$-divisor 
$D$ on 
$A$. Let 
$C$ be a curve in 
$F$. Then
$$\begin{eqnarray}\begin{array}{@{}rcl@{}}-K_{X}\cdot C\; & =\; & -(K_{A}+D)\cdot C+E_{1}\cdot C+\cdots +E_{\unicode[STIX]{x1D6FC}}\cdot C\\ \; & {<}\; & -(K_{A}+D)\cdot C=-(K_{F}+D|_{F})\cdot C\leqslant -K_{F}\cdot C.\end{array}\end{eqnarray}$$ We note that 
$D|_{F}$ is effective and 
$K_{A}|_{F}=K_{F}$ holds since 
$F$ is a general fiber of 
$A\rightarrow B$. We also note that 
$D|_{F}\cdot C\geqslant 0$ since 
$\unicode[STIX]{x1D70C}(F)=1$. By [Reference FujinoF1, Proposition 2.9] (see also 3.1.8), there exists a torus invariant curve 
$C$ on 
$F$ such that 
$-K_{F}\cdot C\leqslant \dim F+1=n-\unicode[STIX]{x1D6FD}+1$. Therefore, we obtain
$$\begin{eqnarray}-K_{X}\cdot C<n-\unicode[STIX]{x1D6FD}+1=d+1\leqslant n\end{eqnarray}$$ since 
$\unicode[STIX]{x1D6FD}\geqslant \unicode[STIX]{x1D6FC}\geqslant 1$. This means that 
$l(R)<d+1$. By combining it with Step 1, we have 
$l(R)<d+1$ without assuming that 
$X$ is 
$\mathbb{Q}$-factorial.
We close this step with easy useful remarks.
Remark 3.2.2. We note that if 
$F\not \simeq \mathbb{P}^{n-\unicode[STIX]{x1D6FD}}$ in the above argument, then we can choose 
$C$ such that 
$-K_{F}\cdot C\leqslant \dim F=n-\unicode[STIX]{x1D6FD}$ (see Theorem 3.1.1).
Remark 3.2.3. If 
$X$ is Gorenstein, then 
$-K_{X}\cdot C<n$ implies 
$-K_{X}\cdot C\leqslant n-1$. Therefore, by combining it with Step 1, we can easily see that the estimate 
$l(R)\leqslant n-1$ always holds for Gorenstein (not necessarily 
$\mathbb{Q}$-factorial) toric varieties.
If 
$\unicode[STIX]{x1D711}_{R}$ is small, then we can find 
$C$ such that 
$-K_{X}\cdot C<n-1$ and 
$[C]\in R$ since we know 
$\unicode[STIX]{x1D6FD}\geqslant \unicode[STIX]{x1D6FC}\geqslant 2$. Therefore, by combining it with Step 1, the estimate 
$l(R)<n-1$ always holds for (not necessarily 
$\mathbb{Q}$-factorial) toric varieties, when 
$\unicode[STIX]{x1D711}_{R}$ is small.
Step 3. In this step, we will investigate the case where 
$l(R)\geqslant n-1$ under the assumption that 
$X$ is 
$\mathbb{Q}$-factorial.
We will use the same notation as in Step 2. In this case, we see that 
$-K_{X}\cdot C\geqslant n-1$ for every curve 
$C$ on 
$F$. Then, we see that 
$\dim A=\dim F=n-1$, 
$F\simeq \mathbb{P}^{n-1}$ and 
$\dim B=0$ (see Remark 3.2.2). Equivalently, 
$\unicode[STIX]{x1D711}_{R}$ contracts 
$F\simeq \mathbb{P}^{n-1}$ to a torus invariant point 
$P\in W$. Let 
$\langle e_{1},\ldots ,e_{n}\rangle$ be the 
$n$-dimensional cone corresponding to 
$P\in W$. Then 
$X$ is obtained by the star subdivision of 
$\langle e_{1},\ldots ,e_{n}\rangle$ by 
$e_{n+1}$, where 
$be_{n+1}=a_{1}e_{1}+\cdots +a_{n}e_{n}$, 
$b\in \mathbb{Z}_{{>}0}$ and 
$a_{i}\in \mathbb{Z}_{{>}0}$ for all 
$i$. We may assume that 
$\gcd (b,a_{1},\ldots ,a_{n})=1$, 
$\gcd (b,a_{1},\ldots ,a_{i-1},a_{i+1},\ldots a_{n})=1$ for all 
$i$, and 
$\gcd (a_{1},\ldots ,a_{n})=1$. Without loss of generality, we may assume that 
$a_{1}\leqslant \cdots \leqslant a_{n}$ by changing the order. We write 
$\unicode[STIX]{x1D70E}_{i}=\langle e_{1},\ldots ,e_{i-1},e_{i+1},\ldots ,e_{n+1}\rangle$ for all 
$i$ and 
$\unicode[STIX]{x1D707}_{k,l}=\unicode[STIX]{x1D70E}_{k}\cap \unicode[STIX]{x1D70E}_{l}$ for 
$k\neq l$. Then
$$\begin{eqnarray}-K_{X}\cdot V(\unicode[STIX]{x1D707}_{k,n})=\frac{1}{a_{n}}\biggl(\mathop{\sum }_{i=1}^{n}a_{i}-b\biggr)\frac{\operatorname{mult}(\unicode[STIX]{x1D707}_{k,n})}{\operatorname{mult}(\unicode[STIX]{x1D70E}_{k})}\geqslant n-1\end{eqnarray}$$ for 
$1\leqslant k\leqslant n-1$. Then 
$\operatorname{mult}(\unicode[STIX]{x1D707}_{k,n})=\operatorname{mult}(\unicode[STIX]{x1D70E}_{k})$ for 
$1\leqslant k\leqslant n-1$. Thus, 
$a_{k}$ divides 
$a_{n}$ for 
$1\leqslant k\leqslant n-1$.
Case 1. If 
$a_{1}=a_{n}$, then 
$a_{1}=\cdots =a_{n}=1$. In this case 
$-K_{X}\cdot V(\unicode[STIX]{x1D707}_{k,n})\geqslant n-1$ implies 
$b=1$. And we have 
$\operatorname{mult}(\unicode[STIX]{x1D707}_{k,l})=\operatorname{mult}(\unicode[STIX]{x1D70E}_{k})$ for 
$1\leqslant k\leqslant n$, 
$1\leqslant l\leqslant n$, and 
$k\neq l$. In particular, 
$\operatorname{mult}(\unicode[STIX]{x1D70E}_{1})=\operatorname{mult}(\unicode[STIX]{x1D707}_{1,l})$ for 
$2\leqslant l\leqslant n$. This implies 
$\operatorname{mult}(\unicode[STIX]{x1D70E}_{1})=1$. Since 
$e_{n+1}=e_{1}+\cdots +e_{n}$, 
$\langle e_{1},\ldots ,e_{n}\rangle$ is a nonsingular cone. Therefore, 
$\unicode[STIX]{x1D711}_{R}:X\rightarrow W$ is a blow-up at a smooth point 
$P$. Of course, 
$l(R)=n-1$.
Case 2. Assume that 
$a_{1}\neq a_{n}$. If 
$a_{2}\neq a_{n}$, then 
$a_{1}/a_{n}\leqslant \frac{1}{2}$ and 
$a_{2}/a_{n}\leqslant \frac{1}{2}$. This contradicts 
$-K_{X}\cdot V(\unicode[STIX]{x1D707}_{k,l})\geqslant n-1$. Therefore, 
$a_{1}=1$ and 
$a_{2}=\cdots =a_{n}=a$ for some positive integer 
$a\geqslant 2$. The condition 
$-K_{X}\cdot V(\unicode[STIX]{x1D707}_{k,n})\geqslant n-1$ implies 
$b=1$. Thus, 
$\operatorname{mult}(\unicode[STIX]{x1D707}_{k,l})=\operatorname{mult}(\unicode[STIX]{x1D70E}_{k})$ for 
$1\leqslant k\leqslant n$, 
$2\leqslant l\leqslant n$, and 
$k\neq l$. In particular, 
$\operatorname{mult}(\unicode[STIX]{x1D70E}_{1})=\operatorname{mult}(\unicode[STIX]{x1D707}_{1,l})$ for 
$2\leqslant l\leqslant n$. Thus, 
$\operatorname{mult}(\unicode[STIX]{x1D70E}_{1})=1$. Since
$$\begin{eqnarray}e_{n+1}=e_{1}+ae_{2}+\cdots +ae_{n},\end{eqnarray}$$
$\langle e_{1},\ldots ,e_{n}\rangle$ is a nonsingular cone. Therefore, 
$\unicode[STIX]{x1D711}_{R}:X\rightarrow W$ is a weighted blow-up at a smooth point 
$P\in W$ with the weight 
$(1,a,\ldots ,a)$. In this case, 
$K_{X}=\unicode[STIX]{x1D711}_{R}^{\ast }K_{W}+(n-1)aE$, where 
$E\simeq \mathbb{P}^{n-1}$ is the exceptional divisor and 
$l(R)=n-1$ (see Proposition 3.2.6).
Anyway, when 
$X$ is 
$\mathbb{Q}$-factorial, we obtain that 
$l(R)\geqslant n-1$ implies 
$l(R)=n-1$. Therefore, the estimate 
$l(R)\leqslant n-1$ always holds when 
$X$ is 
$\mathbb{Q}$-factorial and 
$\unicode[STIX]{x1D711}_{R}$ is birational.
Step 4. In this final step, we will treat the case where 
$l(R)\geqslant n-1$ under the assumption that 
$X$ is not necessarily 
$\mathbb{Q}$-factorial.
Let 
$\unicode[STIX]{x1D70B}:\widetilde{X}\rightarrow X$ be a small projective 
$\mathbb{Q}$-factorialization as in Step 1. By the argument in Step 1, we can find a 
$K_{\widetilde{X}}$-negative extremal ray 
$\widetilde{R}$ of 
$\operatorname{NE}(\widetilde{X}/W)$ such that 
$l(\widetilde{R})\geqslant n-1$. Therefore, by Step 3, the associated contraction morphism 
$\unicode[STIX]{x1D711}_{\widetilde{R}}:\widetilde{X}\rightarrow \widetilde{W}$ is a weighted blow-up at a smooth point 
$\widetilde{P}\in \widetilde{W}$ with the weight 
$(1,a,\ldots ,a)$ for some positive integer 
$a$. Let 
$\widetilde{E}$
$(\simeq \mathbb{P}^{n-1})$ be the 
$\unicode[STIX]{x1D711}_{\widetilde{R}}$-exceptional divisor on 
$\widetilde{X}$. We put 
$E=\unicode[STIX]{x1D70B}(\widetilde{E})$. Then it is easy to see that 
$E\simeq \mathbb{P}^{n-1}$ and that 
$\unicode[STIX]{x1D70B}:\widetilde{E}\rightarrow E$ is an isomorphism.
Lemma 3.2.4. 
$\unicode[STIX]{x1D70B}:\widetilde{X}\rightarrow X$ is an isomorphism over some open neighborhood of 
$E$.
Proof of Lemma 3.2.4.
We will get a contradiction by assuming that 
$\unicode[STIX]{x1D70B}:\widetilde{X}\rightarrow X$ is not an isomorphism over any open neighborhood of 
$E$. Since 
$\unicode[STIX]{x1D711}_{\widetilde{R}}$ is a weighted blow-up as described in the case where 
$X$ is 
$\mathbb{Q}$-factorial (see Step 3) and 
$\unicode[STIX]{x1D70B}$ is a crepant small toric morphism by construction, the fan of 
$\widetilde{X}$ contains 
$n$-dimensional cones
$$\begin{eqnarray}\unicode[STIX]{x1D70E}_{i}:=\langle \{e_{1},\ldots ,e_{n+1}\}\setminus \{e_{i}\}\rangle ,\end{eqnarray}$$ for 
$1\leqslant i\leqslant n$, where 
$\{e_{1},\ldots ,e_{n}\}$ is the standard basis of 
$N=\mathbb{Z}^{n}$ and 
$e_{n+1}:=e_{1}+ae_{2}+\cdots +ae_{n}$ with 
$a\in \mathbb{Z}_{{>}0}$. Since we assume that 
$\unicode[STIX]{x1D70B}:\widetilde{X}\rightarrow X$ is not an isomorphism over any open neighborhood of 
$E$, there exists at least one nonsimplicial 
$n$-dimensional cone 
$\unicode[STIX]{x1D70E}$ in the fan of 
$X$ such that 
$\unicode[STIX]{x1D70E}$ contains one of the above 
$n$-dimensional cones. By symmetry, it is sufficient to consider the two cases where 
$\unicode[STIX]{x1D70E}$ contains 
$\unicode[STIX]{x1D70E}_{n}$ or 
$\unicode[STIX]{x1D70E}_{1}$.
First, we suppose 
$\unicode[STIX]{x1D70E}_{n}\subset \unicode[STIX]{x1D70E}$. Let 
$x=x_{1}e_{1}+\cdots +x_{n}e_{n}\in N$ be the primitive generator for some one-dimensional face of 
$\unicode[STIX]{x1D70E}$ which is not contained in 
$\unicode[STIX]{x1D70E}_{n}$. Then, by considering the facets of 
$\unicode[STIX]{x1D70E}_{n}$, we have the inequalities 
$ax_{1}-x_{n}\geqslant 0$, 
$x_{i}-x_{n}\geqslant 0$ for 
$2\leqslant i\leqslant n-1$, and 
$x_{n}<0$. If 
$x_{1}-x_{n}<0$, then 
$x_{1}<x_{n}<0$. This means that 
$ax_{1}-x_{n}\leqslant x_{1}-x_{n}<0$. This is a contradiction. Therefore, the inequality 
$x_{1}-x_{n}\geqslant 0$ also holds.
Claim. 
$x_{i}\leqslant 0$ for every 
$i\neq n$.
Proof of Claim.
Suppose 
$x_{i}>0$ for some 
$i\neq n$. We note that 
$x$ must be contained in the hyperplane passing through the points 
$e_{1},\ldots ,e_{n-1},e_{n+1}$ since 
$\unicode[STIX]{x1D70B}$ is crepant, that is, 
$K_{\widetilde{X}}=\unicode[STIX]{x1D70B}^{\ast }K_{X}$. So the equality
$$\begin{eqnarray}\displaystyle 1 & = & \displaystyle x_{1}+\cdots +x_{n-1}-(n-2)x_{n}\nonumber\\ \displaystyle & = & \displaystyle (x_{1}-x_{n})+\cdots +(x_{i-1}-x_{n})+x_{i}+(x_{i+1}-x_{n})+\cdots +(x_{n-1}-x_{n})\nonumber\end{eqnarray}$$ holds. Therefore, 
$x_{j}-x_{n}=0$ must hold for every 
$j\neq i$, and 
$x_{i}=1$. If 
$i\neq 1$, then we have 
$a=1$ since 
$ax_{1}-x_{n}=(a-1)x_{n}\geqslant 0$ and 
$x_{n}<0$. However, the linear relation
$$\begin{eqnarray}x+(-x_{n})e_{n+1}=(1-x_{n})e_{i}\end{eqnarray}$$ means that 
$\unicode[STIX]{x1D70B}$ contracts a divisor 
$V(e_{i})$. This is a contradiction because 
$\unicode[STIX]{x1D70B}$ is small by construction. If 
$i=1$, then we have 
$ax_{1}-x_{n}=a-x_{n}>0$ since 
$a>0$ and 
$-x_{n}>0$. However, the linear relation
$$\begin{eqnarray}ax+(-x_{n})e_{n+1}=(a-x_{n})e_{1}\end{eqnarray}$$ means that 
$\unicode[STIX]{x1D70B}$ contracts a divisor 
$V(e_{1})$. This is a contradiction because 
$\unicode[STIX]{x1D70B}$ is small by construction. In any case, we obtain that 
$x_{i}\leqslant 0$ holds for 
$1\leqslant i\leqslant n-1$.◻
Therefore, the linear relation
$$\begin{eqnarray}(-x_{1})e_{1}+\cdots +(-x_{n})e_{n}+x=0\end{eqnarray}$$ says that the cone 
$\langle e_{1},\ldots ,e_{n},x\rangle$ contains a positive dimensional linear subspace of 
$N_{\mathbb{R}}$ because 
$-x_{i}\geqslant 0$ for 
$1\leqslant i\leqslant n-1$ and 
$-x_{n}>0$. Since 
$\langle e_{1},\ldots ,e_{n},x\rangle$ must be contained in a strongly convex cone in the fan of 
$W$, this is a contradiction.
Next, we suppose 
$\unicode[STIX]{x1D70E}_{1}\subset \unicode[STIX]{x1D70E}$. We can apply the same argument as above. Let 
$x=x_{1}e_{1}+\cdots +x_{n}e_{n}\in N$ be the primitive generator for some one-dimensional face of 
$\unicode[STIX]{x1D70E}$ which is not contained in 
$\unicode[STIX]{x1D70E}_{1}$. In this case, we can obtain the inequalities 
$x_{i}-ax_{1}\geqslant 0$ for 
$2\leqslant i\leqslant n$, and 
$x_{1}<0$ by considering the facets of 
$\unicode[STIX]{x1D70E}_{1}$, and the equality 
$(1-(n-1)a)x_{1}+x_{2}+\cdots +x_{n}=1$ by the fact that 
$\unicode[STIX]{x1D70B}$ is crepant. If 
$x_{i}>0$ for some 
$2\leqslant i\leqslant n$, then the equality
$$\begin{eqnarray}\displaystyle 1 & = & \displaystyle (1-(n-1)a)x_{1}+x_{2}+\cdots +x_{n}\nonumber\\ \displaystyle & = & \displaystyle (1-a)x_{1}+(x_{2}-ax_{1})+\cdots +(x_{i-1}-ax_{1})+x_{i}\nonumber\\ \displaystyle & & \displaystyle +\,(x_{i+1}-ax_{1})+\cdots +(x_{n}-ax_{1})\nonumber\end{eqnarray}$$ tells us that 
$a=1$ because 
$x_{1}<0$, and that 
$x_{j}-x_{1}=0$ for every 
$j\neq i$ and 
$x_{i}=1$. Therefore, as in the proof of claim, we get a contradiction by the linear relation
$$\begin{eqnarray}x+(-x_{1})e_{n+1}=(1-x_{1})e_{i}.\end{eqnarray}$$ So we obtain that 
$x_{i}\leqslant 0$ holds for 
$2\leqslant i\leqslant n$. Thus we get a linear relation
$$\begin{eqnarray}(-x_{1})e_{1}+\cdots +(-x_{n})e_{n}+x=0\end{eqnarray}$$ as above, where 
$-x_{i}\geqslant 0$ for 
$2\leqslant i\leqslant n$ and 
$-x_{1}>0$. This means that the cone 
$\langle e_{1},\ldots ,e_{n},x\rangle$ contains a positive dimensional linear subspace of 
$N_{\mathbb{R}}$. This is a contradiction as explained above.
In any case, we get a contradiction. Therefore, 
$\unicode[STIX]{x1D70B}:\widetilde{X}\rightarrow X$ is an isomorphism over some open neighborhood of 
$E$.◻
Since 
$\unicode[STIX]{x1D70B}:\widetilde{X}\rightarrow X$ is an isomorphism over some open neighborhood of 
$E$ by Lemma 3.2.4, we see that 
$E$ is 
$\mathbb{Q}$-Cartier. Therefore, the exceptional locus of 
$\unicode[STIX]{x1D711}_{R}$ coincides with 
$E\simeq \mathbb{P}^{n-1}$. Thus 
$\unicode[STIX]{x1D711}_{R}:X\rightarrow W$ is a weighted blow-up at a torus invariant smooth point 
$P\in W$ with the weight 
$(1,a,\ldots ,a)$ for some positive integer 
$a$.
So, we complete the proof of Theorem 3.2.1. ◻
Remark 3.2.5. If 
$B$ is complete, then we can make 
$C$ a torus invariant curve on 
$X$ in Theorem 3.2.1. For the details, see the proof of [Reference FujinoF1, Theorem 0.1].
We explicitly state the basic properties of the weighted blow-up in Theorem 3.2.1 for the reader’s convenience.
Proposition 3.2.6. Let 
$\unicode[STIX]{x1D711}:X\rightarrow \mathbb{A}^{n}$ be the weighted blow-up at 
$0\in \mathbb{A}^{n}$ with the weight 
$(1,a,\ldots ,a)$ for some positive integer 
$a$. If 
$a=1$, then 
$\unicode[STIX]{x1D711}$ is the standard blow-up at 
$0$. In particular, 
$X$ is smooth. If 
$a\geqslant 2$, then 
$X$ has only canonical Gorenstein singularities which are not terminal singularities. Furthermore, the exceptional locus 
$E$ of 
$\unicode[STIX]{x1D711}$ is isomorphic to 
$\mathbb{P}(1,a,\ldots ,a)\simeq \mathbb{P}^{n-1}$ and
$$\begin{eqnarray}K_{X}=\unicode[STIX]{x1D711}^{\ast }K_{\mathbb{ A}^{n}}+(n-1)aE.\end{eqnarray}$$ We note that 
$E$ is not Cartier on 
$X$ if 
$a\neq 1$. However, 
$aE$ is a Cartier divisor on 
$X$.
Proof. We can check the statements by direct calculation. ◻
Let us see an important related example.
Example 3.2.7. We fix 
$N=\mathbb{Z}^{n}$ and let 
$\{e_{1},\ldots ,e_{n}\}$ be the standard basis of 
$N$. We consider the cone 
$\unicode[STIX]{x1D70E}=\langle e_{1},\ldots ,e_{n}\rangle$ in 
$N^{\prime }=N+\mathbb{Z}e_{n+1}$, where 
$e_{n+1}=(1/b)(1,a,\ldots ,a)$. Here, 
$a$ and 
$b$ are positive integers such that 
$\gcd (a,b)=1$. We put 
$Y=X(\unicode[STIX]{x1D70E})$ is the associated affine toric variety which has only one singular point 
$P$. We take a weighted blow-up of 
$Y$ at 
$P$ with the weight 
$(1/b)(1,a,\ldots ,a)$. This means that we divide 
$\unicode[STIX]{x1D70E}$ by 
$e_{n+1}$ and obtain a fan 
$\unicode[STIX]{x1D6F4}$ of 
$N_{\mathbb{R}}^{\prime }$. We define 
$X=X(\unicode[STIX]{x1D6F4})$. Then the induced projective birational toric morphism 
$f:X\rightarrow Y$ is the desired weighted blow-up. It is obvious that 
$X$ is 
$\mathbb{Q}$-factorial and 
$\unicode[STIX]{x1D70C}(X/Y)=1$. We can easily obtain
$$\begin{eqnarray}K_{X}=f^{\ast }K_{Y}+\biggl(\frac{1+(n-1)a}{b}-1\biggr)E,\end{eqnarray}$$ where 
$E=V(e_{n+1})\simeq \mathbb{P}^{n-1}$ is the exceptional divisor of 
$f$, and
$$\begin{eqnarray}-K_{X}\cdot C=(n-1)-\frac{b-1}{a},\end{eqnarray}$$ where 
$C=V(\langle e_{2},\ldots ,e_{n-1},e_{n+1}\rangle )$ is a torus invariant irreducible curve on 
$E$. We note that
$$\begin{eqnarray}-(K_{X}+\unicode[STIX]{x1D6FF}E)\cdot C>n-1\end{eqnarray}$$if and only if
$$\begin{eqnarray}\unicode[STIX]{x1D6FF}>\frac{b-1}{b}\end{eqnarray}$$ since 
$E\cdot C=-b/a$.
In Section 3.3, we will see more sophisticated examples (see Examples 3.3.1 and 3.3.2), which show the estimates obtained in Theorem 3.2.1 are the best.
By the proof of Theorem 3.2.1, we can prove the following theorem.
Theorem 3.2.8. Let 
$f:X\rightarrow Y$ be a projective toric morphism with 
$\dim X=n$ and let 
$\unicode[STIX]{x1D6E5}=\sum \unicode[STIX]{x1D6FF}_{i}\unicode[STIX]{x1D6E5}_{i}$ be an effective 
$\mathbb{R}$-divisor on 
$X$, where 
$\unicode[STIX]{x1D6E5}_{i}$ is a torus invariant prime divisor and 
$0\leqslant \unicode[STIX]{x1D6FF}_{i}\leqslant 1$ for every 
$i$. Let 
$R$ be an extremal ray of 
$\operatorname{NE}(X/Y)$ and let 
$\unicode[STIX]{x1D711}_{R}:X\rightarrow W$ be the extremal contraction morphism associated to 
$R$. Assume that 
$X$ is 
$\mathbb{Q}$-factorial and 
$\unicode[STIX]{x1D711}_{R}$ is birational. If
$$\begin{eqnarray}\min _{[C]\in R}(-(K_{X}+\unicode[STIX]{x1D6E5})\cdot C)>n-1,\end{eqnarray}$$ then 
$\unicode[STIX]{x1D711}_{R}:X\rightarrow W$ is the weighted blow-up described in Example 3.2.7 and 
$\operatorname{Supp}\unicode[STIX]{x1D6E5}\supset E$, where 
$E\simeq \mathbb{P}^{n-1}$ is the exceptional divisor of 
$\unicode[STIX]{x1D711}_{R}$.
Proof. We use the same notation as in Step 2 in the proof of Theorem 3.2.1. Since
$$\begin{eqnarray}(E_{1}+\cdots +E_{\unicode[STIX]{x1D6FC}}-\unicode[STIX]{x1D6E5})\cdot C\leqslant 0,\end{eqnarray}$$we obtain
$$\begin{eqnarray}\displaystyle -(K_{X}+\unicode[STIX]{x1D6E5})\cdot C & = & \displaystyle -(K_{A}+D)\cdot C+(E_{1}+\cdots +E_{\unicode[STIX]{x1D6FC}}-\unicode[STIX]{x1D6E5})\cdot C\nonumber\\ \displaystyle & {\leqslant} & \displaystyle -(K_{A}+D)\cdot C\nonumber\\ \displaystyle & {\leqslant} & \displaystyle -K_{F}\cdot C\nonumber\end{eqnarray}$$ (see (3.1)). By assumption, 
$-(K_{X}+\unicode[STIX]{x1D6E5})\cdot C>n-1$. This implies that 
$n-1<-K_{F}\cdot C$. Therefore, we obtain 
$\dim A=\dim F=n-1$, 
$F\simeq \mathbb{P}^{n-1}$ and 
$\dim B=0$. In this situation,
$$\begin{eqnarray}-(K_{X}+\unicode[STIX]{x1D6E5})\cdot C\leqslant -(K_{X}+A)\cdot C\end{eqnarray}$$always holds. Thus we have
$$\begin{eqnarray}-(K_{X}+A)\cdot V(\unicode[STIX]{x1D707}_{k,n})=\frac{1}{a_{n}}\biggl(\mathop{\sum }_{i=1}^{n}a_{i}\biggr)\frac{\operatorname{mult}(\unicode[STIX]{x1D707}_{k,n})}{\operatorname{mult}(\unicode[STIX]{x1D70E}_{k})}>n-1\end{eqnarray}$$ for 
$1\leqslant k\leqslant n-1$ (see (3.2)). We note that 
$A=V(e_{n+1})$. Thus, by the same arguments as in the proof of Theorem 3.2.1, we see that 
$\unicode[STIX]{x1D711}_{R}$ is the weighted blow-up described in Example 3.2.7. More precisely, we obtain that 
$(a_{1},\ldots ,a_{n})=(1,\ldots ,1)$ or 
$(1,a,\ldots ,a)$ and that 
$\unicode[STIX]{x1D70E}_{1}$ is a nonsingular cone. However, 
$b$ is not necessarily 
$1$ in the proof of Theorem 3.2.1. By direct calculation, we have 
$\operatorname{Supp}\unicode[STIX]{x1D6E5}\supset E$, where 
$E(=A=F)$ is the exceptional divisor of 
$\unicode[STIX]{x1D711}_{R}$.◻
Finally, we prove the following theorem.
Theorem 3.2.9. (Theorem 1.3)
Let 
$X$ be a 
$\mathbb{Q}$-Gorenstein projective toric 
$n$-fold with 
$\unicode[STIX]{x1D70C}(X)\geqslant 2$. Let 
$R$ be a 
$K_{X}$-negative extremal ray of 
$\operatorname{NE}(X)$ such that
$$\begin{eqnarray}l(R)=\min _{[C]\in R}(-K_{X}\cdot C)>n-1.\end{eqnarray}$$ Then the extremal contraction 
$\unicode[STIX]{x1D711}_{R}:X\rightarrow W$ associated to 
$R$ is a 
$\mathbb{P}^{n-1}$-bundle over 
$\mathbb{P}^{1}$.
Proof. We divide the proof into several steps. From Step 1 to Step 4, we will prove this theorem under the extra assumption that 
$X$ is 
$\mathbb{Q}$-factorial. In Step 5, we will prove that 
$X$ is always 
$\mathbb{Q}$-factorial if there exists an extremal ray 
$R$ with 
$l(R)>n-1$.
Step 1. We consider the contraction morphism 
$\unicode[STIX]{x1D711}_{R}:X\rightarrow W$ associated to 
$R$. By Theorem 3.2.1, 
$\unicode[STIX]{x1D711}_{R}$ is a Fano contraction, that is, 
$\dim W<\dim X$. Let 
$F$ be a general fiber of 
$\unicode[STIX]{x1D711}_{R}$ and let 
$C$ be a curve on 
$F$. Then, by adjunction, we have
$$\begin{eqnarray}-K_{X}\cdot C=-K_{F}\cdot C.\end{eqnarray}$$ We note that 
$F$ is a fake weighted projective space. By Theorem 3.1.1, 
$F\simeq \mathbb{P}^{n-1}$, 
$W=\mathbb{P}^{1}$, and 
$\unicode[STIX]{x1D70C}(X)=2$.
Step 2. Without loss of generality, 
$\unicode[STIX]{x1D711}_{R}:X\rightarrow W$ is induced by the projection 
$\unicode[STIX]{x1D70B}:N=\mathbb{Z}^{n}\rightarrow \mathbb{Z},(x_{1},\ldots ,x_{n})\mapsto x_{n}$. We put
$$\begin{eqnarray}\displaystyle & \displaystyle v_{1}=(1,0,\ldots ,0),\qquad v_{2}=(0,1,0,\ldots ,0),\ldots , & \displaystyle \nonumber\\ \displaystyle & \displaystyle v_{n-1}=(0,\ldots ,0,1,0),\qquad v_{n}=(-1,\ldots ,-1,0), & \displaystyle \nonumber\\ \displaystyle & \displaystyle v_{+}=(b_{1},\ldots ,b_{n-1},a_{+}),\qquad v_{-}=(c_{1},\ldots ,c_{n-1},-a_{-}), & \displaystyle \nonumber\end{eqnarray}$$ where 
$a_{+}$ and 
$a_{-}$ are positive integers. More precisely, 
$v_{i}$ denotes the vector with a 
$1$ in the 
$i$th coordinate and 
$0$’s elsewhere for 
$1\leqslant i\leqslant n-1$. We may assume that the fan 
$\unicode[STIX]{x1D6F4}$ corresponding to the toric variety 
$X$ is the subdivision of 
$N_{\mathbb{R}}$ by 
$v_{1},\ldots ,v_{n},v_{+}$, and 
$v_{-}$. We note that the following equalities
$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}D_{1}-D_{n}+b_{1}D_{+}+c_{1}D_{-}=0\quad \\ D_{2}-D_{n}+b_{2}D_{+}+c_{2}D_{-}=0\quad \\ \qquad \qquad \vdots \quad \\ D_{n-1}-D_{n}+b_{n-1}D_{+}+c_{n-1}D_{-}=0\quad \\ a_{+}D_{+}-a_{-}D_{-}=0\quad \end{array}\right.\end{eqnarray}$$ hold, where 
$D_{i}=V(v_{i})$ for every 
$i$ and 
$D_{\pm }=V(v_{\pm })$. We note that it is sufficient to prove that 
$a_{+}=a_{-}=1$.
Step 3. In this step, we will prove that 
$a_{+}=1$ holds.
By taking a suitable coordinate change, we may assume that
$$\begin{eqnarray}0\leqslant b_{1},\ldots ,b_{n-1}<a_{+}\end{eqnarray}$$ holds. If 
$b_{i}=0$ for every 
$i$, then 
$a_{+}=1$ since 
$v_{+}$ is a primitive vector of 
$N$. From now on, we assume that 
$b_{i_{0}}\neq 0$ for some 
$i_{0}$. Without loss of generality, we may assume that 
$b_{1}\neq 0$. We put
$$\begin{eqnarray}C=V(\langle v_{2},\ldots ,v_{n-1},v_{+}\rangle ).\end{eqnarray}$$ Then 
$C$ is a torus invariant curve contained in a fiber of 
$\unicode[STIX]{x1D711}_{R}:X\rightarrow W$. We have
$$\begin{eqnarray}D_{1}\cdot C=\frac{\operatorname{mult}(\langle v_{2},\ldots ,v_{n-1},v_{+}\rangle )}{\operatorname{mult}(\langle v_{1},\ldots ,v_{n-1},v_{+}\rangle )}=\frac{\gcd (a_{+},b_{1})}{a_{+}}\end{eqnarray}$$ (see 2.1.5) and 
$D_{+}\cdot C=D_{-}\cdot C=0$. We note that 
$D_{i}\cdot C=D_{1}\cdot C$ for every 
$i$ by (3.3). Therefore, we obtain
$$\begin{eqnarray}-K_{X}\cdot C=\frac{n\gcd (a_{+},b_{1})}{a_{+}}.\end{eqnarray}$$ Since 
$0<b_{1}<a_{+}$, we see 
$\gcd (a_{+},b_{1})<a_{+}$. Thus, the following inequality
$$\begin{eqnarray}\frac{\gcd (a_{+},b_{1})}{a_{+}}\leqslant \frac{1}{2}\end{eqnarray}$$holds. This means that
$$\begin{eqnarray}-K_{X}\cdot C\leqslant \frac{n}{2}\leqslant n-1.\end{eqnarray}$$ This is a contradiction. Therefore, 
$b_{i}=0$ for every 
$i$ and 
$a_{+}=1$.
Step 4. By the same argument, we get 
$a_{-}=1$. Thus, we see that 
$\unicode[STIX]{x1D711}_{R}:X\rightarrow W$ is a 
$\mathbb{P}^{n-1}$-bundle over 
$\mathbb{P}^{1}$.
Step 5. In this step, we will prove that 
$X$ is 
$\mathbb{Q}$-factorial.
We assume that 
$X$ is not 
$\mathbb{Q}$-factorial. Let 
$\unicode[STIX]{x1D70B}:\widetilde{X}\rightarrow X$ be a small projective 
$\mathbb{Q}$-factorialization. We note that 
$\unicode[STIX]{x1D70C}(\widetilde{X})>\unicode[STIX]{x1D70C}(X)\geqslant 2$ since 
$X$ is not 
$\mathbb{Q}$-factorial. By the argument in Step 1 in the proof of Theorem 3.2.1, there exists an extremal ray 
$\widetilde{R}$ of 
$\operatorname{NE}(\widetilde{X}/W)$ with 
$l(\widetilde{R})>n-1$. Let 
$\unicode[STIX]{x1D711}_{\widetilde{R}}:\widetilde{X}\rightarrow \widetilde{W}$ be the contraction morphism associated to 
$\widetilde{R}$. Then, by the argument in Step 1, we see that 
$\unicode[STIX]{x1D70C}(\widetilde{X})=2$ and that 
$\unicode[STIX]{x1D711}_{\widetilde{R}}$ is nothing but 
$\unicode[STIX]{x1D711}_{R}\circ \unicode[STIX]{x1D70B}$. This is a contradiction because 
$\unicode[STIX]{x1D70C}(\widetilde{X})>2$. This means that 
$X$ is always 
$\mathbb{Q}$-factorial.
Therefore, we get the desired statement. ◻
We close this subsection with an easy example, which shows that Theorem 3.2.9 is sharp.
Example 3.2.10. We consider 
$N=\mathbb{Z}^{2}$, 
$v_{1}=(0,1),v_{2}=(0,-1),v_{+}=(2,1),v_{-}=(-1,0)$ and the projection 
$\unicode[STIX]{x1D70B}:N=\mathbb{Z}^{2}\rightarrow \mathbb{Z},(x_{1},x_{2})\mapsto x_{1}$. Let 
$\unicode[STIX]{x1D6F4}$ be the fan obtained by subdividing 
$N_{\mathbb{R}}$ by 
$\{v_{1},v_{2},v_{+},v_{-}\}$. Then 
$X=X(\unicode[STIX]{x1D6F4})$ is a projective toric surface with 
$\unicode[STIX]{x1D70C}(X)=2$. The map 
$\unicode[STIX]{x1D70B}:N\rightarrow \mathbb{Z}$ induces a Fano contraction morphism 
$\unicode[STIX]{x1D711}:X\rightarrow \mathbb{P}^{1}$. Let 
$R$ be the corresponding extremal ray of 
$\operatorname{NE}(X)$. Then 
$l(R)=1=2-1$. Note that 
$X$ is not a 
$\mathbb{P}^{1}$-bundle over 
$\mathbb{P}^{1}$.
The reader can find a generalization of Theorem 3.2.9, that is, a characterization of toric projective bundles for 
$\mathbb{Q}$-factorial projective toric varieties, in [Reference Fujino and SatoFS3].
3.3 Examples
In this subsection, we will see that the estimates in Theorem 3.2.1 are the best by the following examples.
Example 3.3.1. We use the same notation as in Example 3.2.7. In Example 3.2.7, we put 
$a=k^{2}$ and 
$b=mk+1$ for any positive integers 
$k$ and 
$m$. Then it is obvious that 
$\gcd (a,b)=1$. So, we can apply the construction in Example 3.2.7. Then we obtain a toric projective birational morphism 
$f:X\rightarrow Y$ such that 
$X$ is 
$\mathbb{Q}$-factorial and 
$\unicode[STIX]{x1D70C}(X/Y)=1$. We can easily check that
$$\begin{eqnarray}K_{X}=f^{\ast }K_{Y}+\biggl(\frac{1+k^{2}(n-1)}{mk+1}-1\biggr)E\end{eqnarray}$$and
$$\begin{eqnarray}-K_{X}\cdot C=n-1-\frac{m}{k}.\end{eqnarray}$$ Therefore, we see that the minimal lengths of extremal rays do not satisfy the ascending chain condition in this birational setting. More precisely, the minimal lengths of extremal rays can take any values in 
$\mathbb{Q}\cap (0,n-1)$. For a related topic, see [Reference Fujino and IshitsukaFI].
Let us construct small contraction morphisms with a long extremal ray.
Example 3.3.2. We fix 
$N=\mathbb{Z}^{n}$ with 
$n\geqslant 3$. Let 
$\{v_{1},\ldots ,v_{n}\}$ be the standard basis of 
$N$. We put
$$\begin{eqnarray}v_{n+1}=(\underbrace{a,\ldots ,a}_{n-k+1},\underbrace{-1,\ldots ,-1}_{k-1})\end{eqnarray}$$ with 
$2\leqslant k\leqslant n-1$, where 
$a$ is any positive integer. Let 
$\unicode[STIX]{x1D6F4}^{+}$ be the fan in 
$\mathbb{R}^{n}$ such that the set of maximal cones of 
$\unicode[STIX]{x1D6F4}^{+}$ is
$$\begin{eqnarray}\{\langle \{v_{1},\ldots ,v_{n+1}\}\setminus \{v_{i}\}\rangle \mid n-k+2\leqslant i\leqslant n+1\}.\end{eqnarray}$$ Let us consider the smooth toric variety 
$X^{+}=X(\unicode[STIX]{x1D6F4}^{+})$ associated to the fan 
$\unicode[STIX]{x1D6F4}^{+}$. We note that the equality
$$\begin{eqnarray}v_{n-k+2}+\cdots +v_{n}+v_{n+1}=av_{1}+\cdots +av_{n-k+1}\end{eqnarray}$$ holds. We can get an antiflipping contraction 
$\unicode[STIX]{x1D711}^{+}:X^{+}\rightarrow W$, that is, a 
$K_{X^{+}}$-positive small contraction morphism, when
$$\begin{eqnarray}a>\frac{k}{n-k+1}.\end{eqnarray}$$In this case, we have the following flipping diagram
By construction, 
$\unicode[STIX]{x1D711}:X\rightarrow W$ is a flipping contraction whose exceptional locus is isomorphic to 
$\mathbb{P}^{n-k}$. The exceptional locus of 
$\unicode[STIX]{x1D711}^{+}$ is isomorphic to 
$\mathbb{P}^{k-1}$. Of course, 
$\unicode[STIX]{x1D711}$ (resp. 
$\unicode[STIX]{x1D711}^{+}$) contracts 
$\mathbb{P}^{n-k}$ (resp. 
$\mathbb{P}^{k-1}$) to a point in 
$W$. We can directly check that
$$\begin{eqnarray}-K_{X}\cdot C=n-k+1-\frac{k}{a}\end{eqnarray}$$ for every torus invariant curve 
$C$ in the 
$\unicode[STIX]{x1D711}$-exceptional locus 
$\mathbb{P}^{n-k}$.
Example 3.3.2 shows that the estimate for small contractions in Theorem 3.2.1 is sharp.
Remark 3.3.3. If 
$(n,k)=(3,2)$ and 
$a\geqslant 2$ in Example 3.3.2, then 
$\unicode[STIX]{x1D711}:X\rightarrow W$ is a threefold toric flipping contraction whose length of the extremal ray is 
${\geqslant}3-2=1$. We note that the lengths of extremal rays of three-dimensional terminal (not necessarily toric) flipping contractions are less than one.
4 Basepoint-free theorems
This section is a supplement to Fujita’s freeness conjecture for toric varieties (see [Reference FujinoF1, Reference FujitaFuj, Reference LaterveerLat, Reference LinLi, Reference MustaţǎMu, Reference PayneP]) and Fulton’s book: [Reference FultonFul].
4.1 Variants of Fujita’s conjectures for toric varieties
One of the most general formulations of Fujita’s freeness conjecture for toric varieties is [Reference FujinoF1, Corollary 0.2]. However, it does not cover the first part of [Reference LinLi, Main theorem A]. So, we give a generalization here with a very simple proof. It is an easy application of the vanishing theorem (see [Reference FujinoF5, Reference FujinoF6]).
Theorem 4.1.1. (Basepoint-freeness)
Let 
$g:Z\rightarrow X$ be a proper toric morphism and let 
$A$ and 
$B$ be reduced torus invariant Weil divisors on 
$Z$ without common irreducible components. Let 
$f:X\rightarrow Y$ be a proper surjective toric morphism and let 
$D$ be an 
$f$-ample Cartier divisor on 
$X$. Then
$$\begin{eqnarray}R^{q}g_{\ast }(\widetilde{\unicode[STIX]{x1D6FA}}_{Z}^{a}(\log (A+B))(-A))\otimes {\mathcal{O}}_{X}(kD)\end{eqnarray}$$ is 
$f$-free, that is,
$$\begin{eqnarray}\displaystyle & & \displaystyle f^{\ast }f_{\ast }(R^{q}g_{\ast }(\widetilde{\unicode[STIX]{x1D6FA}}_{Z}^{a}(\log (A+B))(-A))\otimes {\mathcal{O}}_{X}(kD))\nonumber\\ \displaystyle & & \displaystyle \quad \rightarrow \,R^{q}g_{\ast }(\widetilde{\unicode[STIX]{x1D6FA}}_{Z}^{a}(\log (A+B))(-A))\otimes {\mathcal{O}}_{X}(kD)\nonumber\end{eqnarray}$$ is surjective for every 
$a\geqslant 0$, 
$q\geqslant 0$, and 
$k\geqslant \max _{y\in Y}\dim f^{-1}(y)+1$.
As a very special case, we can recover the following result.
Corollary 4.1.2. (cf. [Reference LinLi, Main theorem A])
Let 
$X$ be an 
$n$-dimensional projective toric variety and let 
$D$ be an ample Cartier divisor on 
$X$. Then the reflexive sheaf 
${\mathcal{O}}_{X}(K_{X}+(n+1)D)$ is generated by its global sections.
Proof. In Theorem 4.1.1, we assume that 
$g:Z\rightarrow X$ is the identity, 
$A=B=0$, 
$a=\dim X$, 
$q=0$, and 
$Y$ is a point. Then we obtain the desired statement.◻
Example 4.1.3. Let us consider 
$X=\mathbb{P}(1,1,1,2)$. Let 
$P$ be the unique 
$\frac{1}{2}(1,1,1)$-singular point of 
$X$ and let 
$D$ be an ample Cartier divisor on 
$X$. We can find a torus invariant curve 
$C$ on 
$X$ such that
$$\begin{eqnarray}K_{X}\cdot C\in {\textstyle \frac{1}{2}}\mathbb{Z}\setminus \mathbb{Z}.\end{eqnarray}$$ Therefore, for every effective Weil divisor 
$E$ on 
$X$ such that 
$E\sim K_{X}+4D$, we have 
$P\in \operatorname{Supp}E$. On the other hand, by Corollary 4.1.2, the reflexive sheaf 
${\mathcal{O}}_{X}(K_{X}+4D)$ is generated by its global sections.
Before proving Theorem 4.1.1, let us recall the definition of the reflexive sheaf 
$\widetilde{\unicode[STIX]{x1D6FA}}_{X}^{a}(\log (A+B))(-A)$ and the vanishing theorem in [Reference FujinoF6].
Definition 4.1.4. Let 
$W$ be any Zariski open set of 
$Z$ such that 
$W$ is smooth and 
$\operatorname{codim}_{Z}(Z\setminus W)\geqslant 2$. In this case, 
$A+B$ is a simple normal crossing divisor on 
$W$. On this assumption, 
$\unicode[STIX]{x1D6FA}_{W}^{a}(\log (A+B))$ is a well-defined locally free sheaf on 
$W$. Let 
$\unicode[STIX]{x1D704}:W{\hookrightarrow}Z$ be the natural open immersion. Then we put
$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D6FA}}_{Z}^{a}(\log (A+B))(-A)=\unicode[STIX]{x1D704}_{\ast }(\unicode[STIX]{x1D6FA}_{W}^{a}(\log (A+B))\otimes {\mathcal{O}}_{W}(-A))\end{eqnarray}$$ for every 
$a\geqslant 0$. It is easy to see that the reflexive sheaf
$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D6FA}}_{Z}^{a}(\log (A+B))(-A)\end{eqnarray}$$ on 
$Z$ does not depend on the choice of 
$W$.
The next theorem is one of the vanishing theorems obtained in [Reference FujinoF6]. For the proof and other vanishing theorems, see [Reference FujinoF5, Reference FujinoF6].
Theorem 4.1.5. [Reference FujinoF6, Theorem 4.3]
Let 
$g:Z\rightarrow X$ be a proper toric morphism and let 
$A$ and 
$B$ be reduced torus invariant Weil divisors on 
$Z$ without common irreducible components. Let 
$f:X\rightarrow Y$ be a proper surjective toric morphism and let 
$L$ be an 
$f$-ample line bundle on 
$X$. Then
$$\begin{eqnarray}R^{p}f_{\ast }(R^{q}g_{\ast }(\widetilde{\unicode[STIX]{x1D6FA}}_{Z}^{a}(\log (A+B))(-A))\otimes L)=0\end{eqnarray}$$ for every 
$p>0$, 
$q\geqslant 0$, and 
$a\geqslant 0$.
Let us prove Theorem 4.1.1.
Proof of Theorem 4.1.1.
By the vanishing theorem: Theorem 4.1.5, we have
$$\begin{eqnarray}R^{p}f_{\ast }(R^{q}g_{\ast }(\widetilde{\unicode[STIX]{x1D6FA}}_{Z}^{a}(\log (A+B))(-A))\otimes {\mathcal{O}}_{X}((k-p)D))=0\end{eqnarray}$$ for every 
$p>0$, 
$q\geqslant 0$, 
$a\geqslant 0$, and 
$k\geqslant \max _{y\in Y}\dim f^{-1}(y)+1$. Since 
${\mathcal{O}}_{X}(D)$ is 
$f$-free, we obtain that
$$\begin{eqnarray}R^{q}g_{\ast }(\widetilde{\unicode[STIX]{x1D6FA}}_{Z}^{a}(\log (A+B))(-A))\otimes {\mathcal{O}}_{X}(kD)\end{eqnarray}$$ is 
$f$-free by the Castelnuovo–Mumford regularity (see, for example, [Reference LazarsfeldLaz, Example 1.8.24]).◻
Here, we treat some generalizations of Fujita’s very ampleness for toric varieties as applications of Theorem 4.1.5. For the details of Fujita’s very ampleness for toric varieties, see [Reference PayneP].
Theorem 4.1.6. Let 
$f:X\rightarrow Y$ be a proper surjective toric morphism, let 
$\unicode[STIX]{x1D6E5}$ be a reduced torus invariant divisor on 
$X$ such that 
$K_{X}+\unicode[STIX]{x1D6E5}$ is Cartier, and let 
$D$ be an 
$f$-ample Cartier divisor on 
$X$. Then 
${\mathcal{O}}_{X}(K_{X}+\unicode[STIX]{x1D6E5}+kD)$ is 
$f$-very ample for every 
$k\geqslant \max _{y\in Y}\dim f^{-1}(y)+2$.
Proof. It follows from the Castelnuovo–Mumford regularity by the vanishing theorem: Theorem 4.1.5. For the details, see [Reference LazarsfeldLaz, Example 1.8.22]. ◻
The following corollary is a special case of the above theorem. This result was first proved by Hui-Wen Lin (see [Reference LinLi]) for the case where 
$X$ is 
$\mathbb{Q}$-factorial and 
$\dim X\leqslant 6$, and then completed by Payne (see [Reference PayneP]).
Corollary 4.1.7. [Reference LinLi, Main Theorem B] and [Reference PayneP, Theorem 1]
Let 
$X$ be an 
$n$-dimensional projective Gorenstein toric variety and let 
$D$ be an ample Cartier divisor on 
$X$. Then 
${\mathcal{O}}_{X}(K_{X}+(n+2)D)$ is very ample.
We think that the following theorem has not been stated explicitly in the literature.
Theorem 4.1.8. (Very ampleness)
Let 
$g:Z\rightarrow X$ be a proper toric morphism and let 
$A$ and 
$B$ be reduced torus invariant Weil divisors on 
$Z$ without common irreducible components. Let 
$f:X\rightarrow Y$ be a proper surjective toric morphism and let 
$D$ be an 
$f$-ample Cartier divisor on 
$X$. Assume that 
$R^{q}g_{\ast }(\widetilde{\unicode[STIX]{x1D6FA}}_{Z}^{a}(\log (A+B))(-A))$ is locally free. Then
$$\begin{eqnarray}R^{q}g_{\ast }(\widetilde{\unicode[STIX]{x1D6FA}}_{Z}^{a}(\log (A+B))(-A))\otimes {\mathcal{O}}_{X}(kD)\end{eqnarray}$$ is 
$f$-very ample for every 
$k\geqslant \max _{y\in Y}\dim f^{-1}(y)+2$.
Proof. The proof of Theorem 4.1.6 works for this theorem since
$$\begin{eqnarray}R^{q}g_{\ast }(\widetilde{\unicode[STIX]{x1D6FA}}_{Z}^{a}(\log (A+B))(-A))\end{eqnarray}$$is locally free by assumption (see [Reference LazarsfeldLaz, Example 1.8.22]). ◻
4.2 Lin’s problem
In this subsection, we treat Lin’s problem raised in [Reference LinLi]. In [Reference LinLi, Lemma 4.3], she claimed the following lemma, which is an exercise in [Reference FultonFul, page 90], without proof.
Lemma 4.2.1. Let 
$X$ be a complete Gorenstein toric variety and let 
$D$ be an ample (Cartier) divisor. If 
$\unicode[STIX]{x1D6E4}(X,K+D)\neq 0$ then 
$K+D$ is generated by its global sections. In fact, 
$P_{K+D}$ is the convex hull of 
$\operatorname{Int}P_{D}\cap M$.
Payne pointed out that Lemma 4.2.1 does not seem to have a known valid proof (see [Reference LinLi, page 500 Added in proof]). Unfortunately, the following elementary example is a counterexample to Lemma 4.2.1. So, Lemma 4.2.1 is NOT true. Therefore, the alternative proof of Theorem A in [Reference LinLi] does not work.
Example 4.2.2. Let 
$Y=\mathbb{P}^{n}$ and let 
$P\in Y$ be a torus invariant closed point. Let 
$f:X\rightarrow Y$ be the blow-up at 
$P$. We put 
$B=\sum _{i=1}^{n+1}B_{i}$, where 
$B_{i}$ is a torus invariant prime divisor on 
$Y$ for every 
$i$. Then it is well known that 
${\mathcal{O}}_{Y}(K_{Y})\simeq {\mathcal{O}}_{Y}(-B)$. We define 
$D=f^{\ast }B-E$, where 
$E$ is the exceptional divisor of 
$f$. In this case, we have
$$\begin{eqnarray}K_{X}=f^{\ast }K_{Y}+(n-1)E\end{eqnarray}$$ and it is not difficult to see that 
$D$ is ample. Therefore,
$$\begin{eqnarray}K_{X}+(n-1)D=f^{\ast }(K_{Y}+(n-1)B)\end{eqnarray}$$ is nef, that is, 
${\mathcal{O}}_{X}(K_{X}+(n-1)D)$ is generated by its global sections. We note that 
$H^{0}(X,{\mathcal{O}}_{X}(K_{X}+aD))\neq 0$ for every positive integer 
$a$. However, 
$K_{X}+aD$ is not nef for any real number 
$a<n-1$. In particular, 
$H^{0}(X,{\mathcal{O}}_{X}(K_{X}+D))\neq 0$ but 
${\mathcal{O}}_{X}(K_{X}+D)$ is not generated by its global sections.
The following theorem is the main theorem of this section. It follows from Theorem 3.2.1.
Theorem 4.2.3. (see Theorem 1.1)
Let 
$X$ be a 
$\mathbb{Q}$-Gorenstein projective toric 
$n$-fold and let 
$D$ be an ample Cartier divisor on 
$X$. Then 
$K_{X}+(n-1)D$ is pseudo-effective if and only if 
$K_{X}+(n-1)D$ is nef.
Proof. If 
$K_{X}+(n-1)D$ is nef, then 
$K_{X}+(n-1)D$ is obviously pseudo-effective. So, all we have to do is to see that 
$K_{X}+(n-1)D$ is nef when it is pseudo-effective. From now on, we assume that 
$K_{X}+(n-1)D$ is pseudo-effective. We take a positive rational number 
$\unicode[STIX]{x1D70F}$ such that 
$K_{X}+\unicode[STIX]{x1D70F}D$ is nef but not ample. In some literature, 
$1/\unicode[STIX]{x1D70F}$ is called the nef threshold of 
$D$with respect to 
$X$. It is not difficult to see that 
$\unicode[STIX]{x1D70F}$ is rational since the Kleiman–Mori cone is a rational polyhedral cone in our case. If 
$\unicode[STIX]{x1D70F}\leqslant n-1$, then the theorem is obvious since
$$\begin{eqnarray}K_{X}+(n-1)D=K_{X}+\unicode[STIX]{x1D70F}D+(n-1-\unicode[STIX]{x1D70F})D\end{eqnarray}$$ and 
$D$ is ample. Therefore, we assume that 
$\unicode[STIX]{x1D70F}>n-1$. We take a sufficiently large positive integer 
$m$ such that 
$m(K_{X}+\unicode[STIX]{x1D70F}D)$ is Cartier. We consider the toric morphism 
$f:=\unicode[STIX]{x1D6F7}_{|m(K_{X}+\unicode[STIX]{x1D70F}D)|}:X\rightarrow Y$. By the definition of 
$\unicode[STIX]{x1D70F}$, 
$f$ is not an isomorphism. Let 
$R$ be an extremal ray of 
$\operatorname{NE}(X/Y)$. Let 
$C$ be any integral curve on 
$X$ such that 
$[C]\in R$. Since 
$(K_{X}+\unicode[STIX]{x1D70F}D)\cdot C=0$, we obtain 
$-K_{X}\cdot C=\unicode[STIX]{x1D70F}D\cdot C>n-1$. Therefore, 
$f$ is not birational by Theorem 3.2.1. Equivalently, 
$K_{X}+\unicode[STIX]{x1D70F}D$ is not big. Thus, the numerical equivalence class of 
$K_{X}+\unicode[STIX]{x1D70F}D$ is on the boundary of the pseudo-effective cone 
$\operatorname{PE}(X)$ of 
$X$. So,
$$\begin{eqnarray}K_{X}+(n-1)D=K_{X}+\unicode[STIX]{x1D70F}D-(\unicode[STIX]{x1D70F}-(n-1))D\end{eqnarray}$$ is outside 
$\operatorname{PE}(X)$. This is a contradiction. Therefore, 
$K_{X}+(n-1)D$ is nef when 
$K_{X}+(n-1)D$ is pseudo-effective.◻
As a corollary, we obtain the following result, which is a correction of Lemma 4.2.1. It is a variant of Fujita’s freeness conjecture for toric varieties. Example 4.2.2 shows that the constant 
$n-1$ in Corollary 4.2.4 is the best.
Corollary 4.2.4. (see Theorem 1.1)
Let 
$X$ be a Gorenstein projective toric 
$n$-fold and let 
$D$ be an ample Cartier divisor on 
$X$. If 
$H^{0}(X,{\mathcal{O}}_{X}(K_{X}+(n-1)D))\neq 0$, then 
${\mathcal{O}}_{X}(K_{X}+(n-1)D)$ is generated by its global sections.
Proof. If 
$H^{0}(X,{\mathcal{O}}_{X}(K_{X}+(n-1)D))\neq 0$, then 
$K_{X}+(n-1)D$ is obviously pseudo-effective. Then, by Theorem 4.2.3, 
$K_{X}+(n-1)D$ is nef. If 
$K_{X}+(n-1)D$ is a nef Cartier divisor, then the complete linear system 
$|K_{X}+(n-1)D|$ is basepoint-free by Lemma 2.2.5.◻
By Theorem 3.2.9, we can check the following result.
Corollary 4.2.5. (Corollary 1.4)
Let 
$X$ be a 
$\mathbb{Q}$-Gorenstein projective toric 
$n$-fold and let 
$D$ be an ample Cartier divisor on 
$X$. If 
$\unicode[STIX]{x1D70C}(X)\geqslant 3$, then 
$K_{X}+(n-1)D$ is always nef. More precisely, if 
$\unicode[STIX]{x1D70C}(X)\geqslant 2$ and 
$X$ is not a 
$\mathbb{P}^{n-1}$-bundle over 
$\mathbb{P}^{1}$, then 
$K_{X}+(n-1)D$ is nef.
Proof. By Theorem 3.2.9, 
$K_{X}+(n-1)D$ is nef since 
$\unicode[STIX]{x1D70C}(X)\geqslant 2$ and 
$X$ is not a 
$\mathbb{P}^{n-1}$-bundle over 
$\mathbb{P}^{1}$.◻
4.3 Supplements to Fujita’s paper
This subsection supplements Fujita’s paper: [Reference FujitaFuj].
We have never seen Corollary 4.2.4 in the literature. However, we believe that Fujita could prove Corollary 4.2.4 without any difficulties (see Theorem 4.3.2). We think that he was not interested in the toric geometry when he wrote [Reference FujitaFuj]. If he was familiar with the toric geometry, then he would have adopted Example 4.3.1 in [Reference FujitaFuj, (3.5) Remark]. This example supplements Fujita’s remark: [Reference FujitaFuj, (3.5) Remark]. We think that our example is much simpler.
Example 4.3.1. We fix 
$N=\mathbb{Z}^{2}$. We put 
$e_{1}=(1,0)$, 
$e_{2}=(0,1)$, 
$e_{3}=(-1,-1)$, and 
$e_{4}=(1,2)$. We consider the fan 
$\unicode[STIX]{x1D6F4}$ obtained by subdividing 
$N_{\mathbb{R}}$ with 
$e_{1}$, 
$e_{2}$, 
$e_{3}$, and 
$e_{4}$. We write 
$X=X(\unicode[STIX]{x1D6F4})$, the associated toric variety. Then 
$X$ is Gorenstein and 
$-K_{X}$ is ample. We put 
$D=-K_{X}$. It is obvious that 
$K_{X}+D\sim 0$. It is easy to see that the Kleiman–Mori cone 
$\operatorname{NE}(X)$ is spanned by the two torus invariant curves 
$E=V(e_{4})$ and 
$E^{\prime }=V(e_{2})$. So, we have two extremal contractions. By removing 
$e_{4}$ from 
$\unicode[STIX]{x1D6F4}$, we obtain a contraction morphism 
$f:X\rightarrow \mathbb{P}^{2}$. In this case, 
$E$ is not Cartier although 
$2E$ is Cartier. We note that 
$-K_{X}\cdot E=1$. The morphism 
$f$ is the weighted blow-up with the weight 
$(1,2)$ described in Proposition 3.2.6. Another contraction is obtained by removing 
$e_{2}$. It is a contraction morphism from 
$X$ to 
$\mathbb{P}(1,1,2)$. Note that 
$E^{\prime }$ is a Cartier divisor on 
$X$.
We close this subsection with the following theorem. In Theorem 4.3.2, we treat normal Gorenstein projective varieties defined over 
$\mathbb{C}$ with only rational singularities, which are not necessarily toric. So, the readers who are interested only in the toric geometry can skip this final theorem.
Theorem 4.3.2. (see [Reference FujitaFuj])
Let 
$X$ be a normal projective variety defined over 
$\mathbb{C}$ with only rational Gorenstein singularities. Let 
$D$ be an ample Cartier divisor on 
$X$. If 
$K_{X}+(n-1)D$ is pseudo-effective with 
$n=\dim X$, then 
$K_{X}+(n-1)D$ is nef.
Proof. We take a positive rational number 
$\unicode[STIX]{x1D70F}$ such that 
$K_{X}+\unicode[STIX]{x1D70F}D$ is nef but not ample. It is well known that 
$\unicode[STIX]{x1D70F}\leqslant n+1$ (see [Reference FujitaFuj, Theorem 1]). If 
$\unicode[STIX]{x1D70F}\leqslant n-1$, then the theorem is obvious. Therefore, we assume that 
$n-1<\unicode[STIX]{x1D70F}\leqslant n+1$. If 
$\unicode[STIX]{x1D70F}=n+1$, then 
$X\simeq \mathbb{P}^{n}$ and 
${\mathcal{O}}_{X}(D)\simeq {\mathcal{O}}_{\mathbb{P}^{n}}(1)$. In this case, 
$K_{X}+(n-1)D$ is not pseudo-effective. Thus, we have 
$n-1<\unicode[STIX]{x1D70F}\leqslant n$ by [Reference FujitaFuj, Theorem 1]. By [Reference FujitaFuj, Theorem 2] and its proof, it can be checked easily that 
$\unicode[STIX]{x1D70F}=n$ and 
$K_{X}+\unicode[STIX]{x1D70F}D=K_{X}+nD$ is nef but is not big. Therefore, 
$K_{X}+nD$ is on the boundary of the pseudo-effective cone of 
$X$. So, 
$K_{X}+(n-1)D=K_{X}+nD-D$ is not pseudo-effective. This is a contradiction. Anyway, we obtain that 
$K_{X}+(n-1)D$ is nef if 
$K_{X}+(n-1)D$ is pseudo-effective.◻
Acknowledgments
The first author was partially supported by JSPS KAKENHI Grant Numbers JP16H03925, JP16H06337. If the first author remembers correctly, he prepared a preliminary version of this paper around 2006 in Nagoya. Then his interests moved to the minimal model program. In 2011, he revised it in Kyoto. The current version was written in Osaka. He thanks the colleagues in Nagoya, Kyoto, and Osaka very much. The authors would like to thank the referee for useful comments.
