On Mori cone of Bott towers

TL;DR

This paper investigates the Mori cone and nef cone of Bott towers, classifies their Fano properties, and proves vanishing theorems for tangent bundle cohomology, advancing understanding of their geometric and algebraic structure.

Contribution

It provides a detailed analysis of the Mori and nef cones of Bott towers, classifies Fano, weak Fano, and log Fano Bott towers, and establishes new vanishing theorems for tangent bundle cohomology.

Findings

Classification of Fano, weak Fano, and log Fano Bott towers.

Explicit description of Mori and nef cones for Bott towers.

Vanishing theorems for tangent bundle cohomology.

Abstract

A Bott tower of height $r$ is a sequence of projective bundles $$X_r \overset{{\pi_r}}\longrightarrow X_{r-1} \overset{\pi_{r-1}}\longrightarrow \cdots \overset{\pi_2}\longrightarrow X_1=\mathbb P^1 \overset{\pi_1} \longrightarrow X_0=\{pt\}, $$ where $X_i=\mathbb P (\mathcal O_{X_{i-1}}\oplus \mathcal L_{i-1})$ for a line bundle $\mathcal L_{i-1}$ over $X_{i-1}$ for all $1\leq i\leq r$ and $\mathbb P(-)$ denotes the projectivization. These are smooth projective toric varieties and we refer to the top object $X_{r}$ also as a Bott tower. In this article, we study the Mori cone and numerically effective (nef) cone of Bott towers, and we classify Fano, weak Fano and log Fano Bott towers. We prove some vanishing theorems for the cohomology of tangent bundle of Bott towers.

Equations

$$X_r\stackrel{{\pi }_r}{⟶}{X}_{r-1}\stackrel{{\pi }_{r-1}}{⟶}\cdots \stackrel{{\pi }_2}{⟶}{X}_1={\mathbb{P}}^1\stackrel{{\pi }_1}{⟶}{X}_0=\{pt\},$$

$$X_r\stackrel{{\pi }_r}{⟶}{X}_{r-1}\stackrel{{\pi }_{r-1}}{⟶}\cdots \stackrel{{\pi }_2}{⟶}{X}_1={\mathbb{P}}^1\stackrel{{\pi }_1}{⟶}\{pt\},$$

$$M_r:={\left[\begin{array}{ccccc}1& {\beta }_{12}& {\beta }_{13}& \dots & {\beta }_{1r}\\ 0& 1& {\beta }_{23}& \dots & {\beta }_{2r}\\ 0& 0& 1& \dots & {\beta }_{3r}\\ ⋮& ⋮& & \ddots & ⋮\\ 0& \dots & \dots & & 1\end{array}\right]}_{r\times r},$$

$${\eta }_i^{+}:=\{r\ge j>i:{\beta }_{ij}>0\}$$

$${\eta }_i^{-}:=\{r\ge j>i:{\beta }_{ij}<0\}.$$

$$⟨-,-⟩:M_{\mathbb{R}}\times N_{\mathbb{R}}\to \mathbb{R}.$$

$$\Sigma (s):=\{\sigma \in \Sigma :dim(\sigma )=s\}.$$

$$\sigma (1):=\Sigma (1)\cap \sigma .$$

$$TDiv(X_{\Sigma })=\underset{\rho \in \Sigma (1)}{⨁}\mathbb{Z}{D}_{\rho }.$$

$$div({\chi }^m)=\sum _{\rho \in \Sigma (1)}⟨m,u_{\rho }⟩D_{\rho }.$$

$$X_r=\mathbb{P}({\mathcal{O}}_{X_{r-1}}\oplus {\mathcal{L}}_{r-1})$$

$$H^{\ast }(X_i,\mathbb{Z})=H^{\ast }(X_{i-1},\mathbb{Z})1\oplus H^{\ast }(X_{i-1},\mathbb{Z})u_i.$$

$$u_i^2=c_1({\mathcal{L}}_{i-1})u_i,$$

$$H^{\ast }(X_i,\mathbb{Z})=H^{\ast }(X_{i-1},\mathbb{Z})[u_i]/J_i,$$

$$0⟶\mathbb{Z}⟶{\mathcal{O}}_{X_{i-1}}⟶{\mathcal{O}}_{X_{i-1}}^{\ast }⟶0.$$

$$c_1({\mathcal{L}}_{i-1})=-\sum _{k=1}^{i-1}{\beta }_{ki}{u}_k\in H^2(X_{i-1},\mathbb{Z}),$$

$$H^{\ast }(X_r,\mathbb{Z})=\mathbb{Z}[u_1,\dots ,u_r]/J,$$

$$M_r:={\left[\begin{array}{ccccc}1& {\beta }_{12}& {\beta }_{13}& \dots & {\beta }_{1r}\\ 0& 1& {\beta }_{23}& \dots & {\beta }_{2r}\\ 0& 0& 1& \dots & {\beta }_{3r}\\ ⋮& ⋮& & \ddots & ⋮\\ 0& \dots & \dots & & 1\end{array}\right]}_{r\times r}$$

$$X_r\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$e_i^{-}:=-e_i^{+}-\sum _{j>i}{\beta }_{ij}{e}_j^{+},$$

$$\{e_j^{ϵ}:1\le j\le i\text{mbox}andϵ\in \{+,-\}\},$$

$$\{e_j^{+},e_j^{-}:1\le j\le i\},$$

$$\Sigma (1)=\{{\rho }_i^{+}:1\le i\le r\}\cup \{{\rho }_i^{-}:1\le i\le r\}.$$

$$0\sim div({\chi }^{e_i^{+}})=\sum _{\rho \in \Sigma (1)}⟨u_{\rho },e_i^{+}⟩D_{\rho },$$

$$X_r\setminus U={\cup }_{i=1}^r{D}_{{\rho }_i^{ϵ}}$$

$$h_i^j:=\{\begin{array}{ll}0& \text{mbox}forj>i.\\ 1& \text{mbox}forj=i.\\ -{\sum }_{k=j}^{i-1}{\beta }_{ik}(h_k^j)& \text{mbox}forj<i.\end{array}$$

$$0\sim div({\chi }^{e_i^{+}})=\sum _{\rho \in \Sigma (1)}⟨u_{\rho },e_i^{+}⟩D_{\rho }.$$

$$e_{i}^{-}=-e_{i}^{+}-\sum_{j>i}\beta_{ij}e_{j}^{+}~{}\mbox{ for all $1\leq i\leq r$}.$$

$$0\sim div({\chi }^{e_1^{+}})=D_{{\rho }_1^{+}}-D_{{\rho }_1^{-}}.$$

$$D_{{\rho }_1^{-}}\sim D_{{\rho }_1^{+}}.$$

Full text

[TABLE]

where $X_{i}=\mathbb{P}(\mathcal{O}_{X_{i-1}}\oplus\mathcal{L}_{i-1})$ for a line bundle $\mathcal{L}_{i-1}$ over $X_{i-1}$ for all $1\leq i\leq r$ and $\mathbb{P}(-)$ denotes the projectivization. These are smooth projective toric varieties and we refer to the top object $X_{r}$ also as a Bott tower. In this article, we study the Mori cone and numerically effective (nef) cone of Bott towers, and we classify Fano, weak Fano and log Fano Bott towers. We prove some vanishing theorems for the cohomology of tangent bundle of Bott towers.

††footnotetext: The author is supported by AGIR Pole MSTIC project run by the University of Grenoble Alpes, France.

Keywords: Bott towers, Mori cone, primitive relations and toric varieties.

1. Introduction

In [BS58], R. Bott and H. Samelson introduced a family of (smooth differentiable) manifolds which may be viewed as the total spaces of iterated $\mathbb{P}^{1}$-bundles over a point $\{pt\}$, where each $\mathbb{P}^{1}$-bundle is the projectivization of a rank $2$ decomposable vector bundle. In [GK94], M. Grossberg and Y. Karshon proved (in complex geometry setting) that these manifolds have a natural action of a compact torus and also obtained some applications to representation theory and symplectic geometry. In [Civ05], Y. Civan proved that these are smooth projective toric varieties. These are called Bott towers, we denote them by $\{(X_{i},\pi_{i}):1\leq i\leq r\}$, where

[TABLE]

$X_{i}=\mathbb{P}(\mathcal{O}_{X_{i-1}}\oplus\mathcal{L}_{i-1})$ for a line bundle $\mathcal{L}_{i-1}$ over $X_{i-1}$ for all $1\leq i\leq r$ and $r$ is the dimension of $X_{r}$. In [CS11], [CMS10] and [Ish12], the authors studied “cohomological rigidity”properties of Bott towers. These also play an important role in algebraic topology and K-theory (see [CR05], [DJ91] and references therein). In this article we refer to $X_{r}$ also as a Bott tower (it is also called Bott manifold).

In this paper we study the geometry of Bott towers in more detail by methods of toric geometry. We work over the field $\mathbb{C}$ of complex numbers. We study the Mori cone of $X_{r}$ and prove that the class of curves corresponding to ‘primitive relations $r(P_{i})$’ forms a basis of the real vector space of numerical classes of one-cycles in $X_{r}$ (see Theorem 4.7 and Corollary 4.8). An extremal ray $R$ in the Mori cone is called Mori ray if $R\cdot K_{X_{r}}<0$, where $K_{X_{r}}$ is the canonical divisor in $X_{r}$. We describe extremal rays and Mori rays of the Mori cone of $X_{r}$ (see Theorem 8.1). We characterize the ampleness and numerically effectiveness of line bundles on $X_{r}$ (see Lemma 5.1) and describe the generators of the nef cone of $X_{r}$ (see Theorem 5.7).

Recall that a smooth projective variety $X$ is called Fano (respectively, weak Fano) if its anti-canonical divisor $-K_{X}$ is ample (respectively, nef and big). Following [AS14], we say that a pair $(X,D)$ of a normal projective variety $X$ and an effective $\mathbb{Q}$-divisor $D$ is log Fano if it is Kawamata log terminal and $-(K_{X}+D)$ is ample (see Section 7 for more details). We study the Fano, weak Fano and the log Fano (of the pair $(X_{r},D)$ for a suitably chosen divisor $D$ in $X_{r}$) properties of the Bott tower $X_{r}$. To describe these results we need some notation. It is known that a Bott tower $\{(X_{i},\pi_{i}):1\leq i\leq r\}$ is uniquely determined by an upper triangular matrix $M_{r}$ with integer entries, defined via the first Chern class of the line bundle $\mathcal{L}_{i-1}$ on $X_{i-1}$, where $X_{i}=\mathbb{P}(\mathcal{O}_{X_{i-1}}\oplus\mathcal{L}_{i-1})$ for $1\leq i\leq r$ (see [GK94, Section 2.3], [Civ05] and [VT15, Section 7.8]). For more details see Section 2. Let

[TABLE]

where $\beta_{ij}$’s are integers. Define for $1\leq i\leq r$,

[TABLE]

and

[TABLE]

If $|\eta_{i}^{-}|=1$ (respectively, $|\eta_{i}^{-}|=2$), then set $\eta^{-}_{i}=\{l\}$ (respectively, $\eta^{-}_{i}=\{l_{1},l_{2}\}$). The following can be viewed as a condition on $i^{th}$ row of the matrix $M_{r}$:

• •

$N^{1}_{i}$ is the condition that

(i) $|\eta^{+}_{i}|=0$, $|\eta^{-}_{i}|\leq 1$, and if $|\eta^{-}_{i}|=1$ then $\beta_{il}=-1$; or

(ii) If $|\eta^{+}_{i}|>0$, then there exists $m>i$ such that $\beta_{im}=1$, $\beta_{ij}=0$ for all $j<m$, and $\beta_{ij}=\beta_{mj}$ for all $j>m$.

• •

$N^{2}_{i}$ is the condition that

(i) Assume that $|\eta^{+}_{i}|=0$. Then $|\eta_{i}^{-}|\leq 2$, and if $|\eta^{-}_{i}|=1(\mbox{respectively,}~{}|\eta^{-}_{i}|=2)$ then $\beta_{il}=-1$ or $-2$ (respectively, $\beta_{il_{1}}=-1=\beta_{il_{2}}$); or

(ii) If $|\eta^{+}_{i}|>0$, then one of the following holds: Let $A_{m,j}:=\beta_{im}\beta_{mj}-\beta_{ij}$.

(a) There exists $m>i$ such that $\beta_{im}=1$ or $2$; $\beta_{ij}=0$ for all $j<m$ and $A_{m,j}=0$ for $j>m$.

(b) There exists $m_{2}>m_{1}>i$ such that $-\beta_{im_{1}}=\beta_{im_{2}}=1$, $\beta_{ij}=0$ for $m_{1}\neq j<m_{2}$ and $A_{m_{2},j}=0$ for $j>m_{2}$.

(c) There exists $m_{2}>m_{1}>i$ such that $\beta_{im_{1}}=A_{m_{1},m_{2}}=1$, $\beta_{ij}=0$ for $j<m_{1}$ and $A_{m_{1},j}=0$ for $m_{1}<j\neq m_{2}$.

(d) There exists $m_{2}>m_{1}>i$ such that $\beta_{im_{1}}=1=-A_{m_{1},m_{2}}$, $\beta_{ij}=0$ for $j<m_{1}$, $A_{m_{1},j}=0$ for $m_{1}<j<m_{2}$ and $\beta_{m_{2}j}+A_{m_{1},j}=0$ for $j>m_{2}$.

Definition 1.1**.**

We say $X_{r}$ satisfies condition $I$ (respectively, condition $II$) if $N^{1}_{i}$ (respectively, $N^{2}_{i}$) holds for all $1\leq i\leq r$.

Note that $N_{i}^{1}\Longrightarrow N_{i}^{2}$ for all $1\leq i\leq r$. If $X_{r}$ satisfies condition $I$, then it also satisfies conditions $II$. We prove,

Theorem** (see Theorem 6.3).**

1. (1)

$X_{r}$* is Fano if and only if it satisfies $I$.* 2. (2)

$X_{r}$* is weak Fano if and only if it satisfies $II$.*

As a consequence we get some vanishing results for the cohomology of tangent bundle of Bott towers and hence local rigidity results. Let $T_{X_{r}}$ denote the tangent bundle of $X_{r}$.

Corollary** (see Corollary 6.5 and Corollary 6.6).**

If $X_{r}$ satisfies $I$, then $H^{i}(X_{r},T_{X_{r}})=0$ for all $i\geq 1$. In particular, $X_{r}$ is locally rigid.

For $1\leq i\leq r$, we define some constants $k_{i}$ which again depend on the given matrix $M_{r}$ corresponding to the Bott tower $X_{r}$ (for more details see Section 7). We prove,

Theorem** (see Theorem 7.1).**

The pair $(X_{r},D)$ is log Fano if and only if $k_{i}<0$ for all $1\leq i\leq r$.

Remark 1.2**.**

By using the results of this article, in [Cha17b] we give some applications to Bott-Samelson-Demazure-Hansen (BSDH) variety, which can be described also as a iterated projective line bundle, by degeneration of this variety to a Bott tower. Precisely, we study Fano, weak Fano, log Fano properties for BSDH varieties (see also [Cha17a]). We obtain some vanishing theorems for the cohomology of tangent bundle (and line bundles) on BSDH varieties (see also [CKP15], [CKP] and [CK17]). We also recover the results in [PK16].

The paper is organized as follows: In Section 2, we discuss preliminaries on Bott towers and toric varieties. In Section 3, we discuss the Picard group of the Bott tower and compute the relative tangent bundle. Section 4 contains detailed study of primitive collections and primitive relations of the Bott tower and we also describe the Mori cone. In Section 5 we describe ample and nef line bundles on the Bott tower, and we find the generators of the nef cone. In Section 6 and 7, we study Fano, weak Fano and log Fan properties for Bott towers. We also see some vanishing results. In Section 8, we describe extremal rays and Mori rays for the Bott tower.

2. Preliminaries

In this section we recall toric varieties (see [CLS11]) and Bott towers (see [Civ05] and [VT15]). We work throughout the article over the field $\mathbb{C}$ of complex numbers. We expect that the proofs work for algebraically closed fields of arbitrary characteristic, but did not find appropriate references in that generality.

2.1. Toric varieties

We briefly recall the structure of toric varieties from [CLS11] (see also [Ful93] and [Oda88]).

Definition 2.1**.**

A normal variety $X$ is called a toric variety (of dimension $n$) if it contains an $n$-dimensional torus $T$ (i.e. $T=(\mathbb{C}^{*})^{n}$) as a Zariski open subset such that the action of the torus on itself by multiplication extends to an action of the torus on $X$.

Toric varieties are completely described by the combinatorics of the corresponding fans. We briefly recall here, let $N$ be the lattice of one-parameter subgroups of $T$ and let $M$ be the lattice of characters of $T$. Let $M_{\mathbb{R}}:=M\otimes\mathbb{R}$ and $N_{\mathbb{R}}:=N\otimes\mathbb{R}$. Then we have a natural bilinear pairing

[TABLE]

A fan $\Sigma$ in $N_{\mathbb{R}}$ is a collection of convex polyhedral cones that is closed under intersections and cone faces. Let $\check{\sigma}$ be the dual cone of $\sigma\in\Sigma$ in $M_{\mathbb{R}}$. For $\sigma\in\Sigma$, the semigroup algebra $\mathbb{C}[\check{\sigma}\cap M]$ is a normal domain and finitely generated $\mathbb{C}$-algebra. Then the scheme $Spec(\mathbb{C}[\check{\sigma}\cap M])$ is called the affine toric variety corresponding to $\sigma$. For a given fan $\Sigma$, we can define a toric variety $X_{\Sigma}$ by gluing the affine toric varieties $Spec(\mathbb{C}[\check{\sigma}\cap M])$ as $\sigma$ varies in $\Sigma$. For all $1\leq s\leq n$,

[TABLE]

For each $\rho\in\Sigma(1)$, we denote $u_{\rho}$, the generator of $\rho\cap N.$ For $\sigma\in\Sigma$,

[TABLE]

There is a bijective correspondence between the cones in $\Sigma$ and the $T$-orbits in $X_{\Sigma}$. For each $\sigma\in\Sigma$, the dimension $dim(O(\sigma))$ of the $T$-orbit $O(\sigma)$ corresponding to $\sigma$ is $n-dim(\sigma)$. Let $\tau,\sigma\in\Sigma$, then $\tau$ is a face of $\sigma$ if and only if $O(\sigma)\subset\overline{O(\tau)}$, where $\overline{O(\sigma)}$ is the closure of $T$-orbit $O(\sigma)$. We denote $V(\sigma)=\overline{O(\sigma)}$ and it is a toric variety with the corresponding fan being Star($\sigma$), the star of $\sigma$ which is the set of cones in $\Sigma$ which have $\sigma$ as a face. Let $D_{\rho}=\overline{O(\rho)}$ be the torus-invariant prime divisor in $X_{\Sigma}$ corresponding to $\rho\in\Sigma(1)$. The group $TDiv(X_{\Sigma})$ of $T$-invariant divisors in $X_{\Sigma}$ is given by

[TABLE]

For each $m\in M$, the character $\chi^{m}$ of $T$ is a rational function on $X_{\Sigma}$ and the corresponding divisor is given by

[TABLE]

2.2. Bott towers

In this section we recall some basic definitions and results on Bott towers. Let $\mathcal{L}_{0}$ be a trivial line bundle over a single point $X_{0}:=\{pt\}$, and let $X_{1}:=\mathbb{P}(\mathcal{O}_{X_{0}}\oplus\mathcal{L}_{0})$, where $\mathbb{P}(-)$ denotes the projectivization. Let $\mathcal{L}_{1}$ be a line bundle on $X_{1}$, then define $X_{2}:=\mathbb{P}(\mathcal{O}_{X_{1}}\oplus\mathcal{L}_{1})$, which is a $\mathbb{P}^{1}$-bundle over $X_{1}$. Repeat this process $r$-times, so that each $X_{i}$ is a $\mathbb{P}^{1}$-bundle over $X_{i-1}$ for $1\leq i\leq r$. We get the following:

[TABLE]

For each $1\leq i\leq r$, $X_{i}$ is a smooth projective toric variety (see [Civ05, Theorem 22]). Consider the points $[1:0]$ and $[0:1]$ in $\mathbb{P}^{1}$, we call them the south pole and the north pole respectively. The zero section of $\mathcal{L}_{i-1}$ gives a section $s^{0}_{i}:X_{i-1}\longrightarrow X_{i}$ , the south pole section; similarly, the north pole section $s^{1}_{i}:X_{i-1}\longrightarrow X_{i}$ by letting the first coordinate in $\mathbb{P}(\mathcal{O}_{X_{i-1}}\oplus\mathcal{L}_{i-1})$ to vanish.

Let $1\leq i\leq r$. Since $\pi_{i}:X_{i}\longrightarrow X_{i-1}$ is a projective bundle, by a standard result on the cohomology ring of projective bundles we have the following (see [Har77, Page 429] for instance, and also [Mil16, Proposition 10.1]):

Theorem 2.2**.**

The cohomology ring $H^{*}(X_{i},\mathbb{Z})$ of $X_{i}$ is a free module over $H^{*}(X_{i-1},\mathbb{Z})$ on generators $1$ and $u_{i}$, which have degree [math] and $2$ respectively, that is

[TABLE]

The ring structure is determined by the single relation

[TABLE]

where $c_{1}(-)$ denotes the first Chern class and the restriction of $u_{i}$ to the fiber $\mathbb{P}^{1}\subset X_{i}$ is the first Chern class of the canonical line bundle over $\mathbb{P}^{1}$. Hence we have

[TABLE]

where $J_{i}$ is the ideal generated by $u_{i}^{2}-c_{1}(\mathcal{L}_{i-1})u_{i}$.

Consider the exponential sequence (see [Har77, Page 446]):

[TABLE]

Then we get the following exact sequence:

$0\longrightarrow H^{1}(X_{i-1},\mathbb{Z})\longrightarrow H^{1}(X_{i-1},\mathcal{O}_{X_{i-1}})\longrightarrow H^{1}(X_{i-1},\mathcal{O}_{X_{i-1}}^{*})\overset{c_{1}(-)}{\longrightarrow}H^{2}(X_{i-1},\mathbb{Z})\longrightarrow H^{2}(X_{i-1},\mathcal{O}_{X_{i-1}})\longrightarrow\cdots$

Since $X_{i-1}$ is toric, we have $H^{j}(X_{i-1},\mathcal{O}_{X_{i-1}})=0$ for all $j>0$ (see [Oda88, Corollary 2.8]). As $H^{1}(X_{i-1},\mathcal{O}_{X_{i-1}}^{*})=Pic(X_{i-1})$, we get $c_{1}(-):Pic(X_{i-1})\overset{\sim}{\longrightarrow}H^{2}(X_{i-1},\mathbb{Z})$. Then we have the following:

Theorem 2.3**.**

Each line bundle $\mathcal{L}_{i-1}$ on $X_{i-1}$ is determined (up to an algebraic isomorphism) by its first Chern class, which can be written as a linear combination

[TABLE]

where $\beta_{ik}$’s are integers for $1\leq k\leq i-1$.

Then by Theorem 2.2 and 2.3, by iteration, we get the following:

Corollary 2.4**.**

We have

[TABLE]

where $J$ is the ideal generated by $\{u_{j}^{2}+\sum_{i<j}\beta_{ij}u_{i}u_{j}:1\leq j\leq r\}$ and the integers $\beta_{ij}$’s are as in Theorem 2.3.

Write $\{\beta_{ij}:1\leq i<j\leq r\}$, the collection of $r(r-1)/2$ integers, as an upper triangular $r\times r$ matrix

[TABLE]

Then we get the following result (see for instance [GK94, Lemma 2.15] and also [Civ05, Section 3]).

Corollary 2.5**.**

There is a bijective correspondence between {Bott towers of height $r$} and {$r\times r$ upper triangular matrices with integer entries as in (2.1)}.

Two Bott towers $\{(X_{i},\pi_{i}):1\leq i\leq r\}$ and $\{(X^{\prime}_{i},\pi^{\prime}_{i}):1\leq i\leq r\}$ are isomorphic if there exists a collection of isomorphisms $\{\phi_{i}:X_{i}\to X_{i}^{\prime}:1\leq i\leq r\}$ such that the following diagram is commutative:

[TABLE]

2.2.1. Toric structure on Bott tower

Let $\{e_{1}^{+},\ldots,e_{r}^{+}\}$ be the standard basis of the lattice $\mathbb{Z}^{r}$. Define, for all $i\in\{1,\ldots,r\}$,

[TABLE]

where $\beta_{ij}$’s are integers as above. Then we have the following theorem (see [Civ05, Section 3 and Theorem 22] and for algebraic topology setting see [VT15, Theorem 7.8.7]):

Theorem 2.6**.**

The Bott tower $\{(X_{i},\pi_{i}):1\leq i\leq r\}$ corresponding to a matrix $M_{r}$ as in (2.1) is isomorphic to $\{(X_{\Sigma_{i}},\pi_{\Sigma_{i}}):1\leq i\leq r\}$, the collection of smooth projective toric varieties corresponding to the fan $\Sigma_{i}$ with the $2^{i}$ maximal cones generated by the set of vectors

[TABLE]

and where $\pi_{\Sigma_{i}}:X_{\Sigma_{i}}\to X_{\Sigma_{i-1}}$ is the toric morphism induced by the projection $\overline{\pi_{\Sigma_{i}}}:\mathbb{Z}^{i}\to\mathbb{Z}^{i-1}$ for all $1\leq i\leq r$.

Note that by Theorem 2.6, $\Sigma_{i}$ has $2i$ one-dimensional cones generated by the vectors

[TABLE]

and by (2.2), we can see that the divisors $D_{\rho_{j}^{+}}$ corresponding to $e_{j}^{+}$ for $1\leq j\leq i$ form a basis of the Picard group of $X_{i}$ (see Section 3 for more details).

3. On Picard group of a Bott tower

Now we describe a basis of the Picard group $Pic(X_{r})$ of $X_{r}$. Let $\epsilon\in\{+,-\}$ and for $1\leq i\leq r$, let $\rho_{i}^{\epsilon}$ be the one-dimensional cone generated by $e_{i}^{\epsilon}$ . For all $1\leq i\leq r$, we define $D_{\rho_{i}^{\epsilon}}$ to be the toric divisor corresponding to the one-dimensional cone $\rho_{i}^{\epsilon}$. We prove,

Lemma 3.1**.**

The set $\{D_{\rho_{i}^{\epsilon}}:1\leq i\leq r~{}\mbox{and}~{}\epsilon\in\{+,-\}\}$ forms a basis of $Pic(X_{r})$.

Proof.

By Theorem 2.6, using the description of the one-dimensional cones we have the following decomposition of $\Sigma(1)$:

[TABLE]

Again by Theorem 2.6, $\{D_{\rho_{i}^{+}}:1\leq i\leq r\}$ forms a basis of the Picard group $Pic(X_{r})$ of $X_{r}$. Since

[TABLE]

by (2.2) we can see that $\{D_{\rho_{i}^{-}}:1\leq i\leq r\}$ also forms a basis of $Pic(X_{r}).$ In general, let $\sigma\in\Sigma$ be the maximal cone generated by $\{e_{i}^{\epsilon}:1\leq i\leq r\}$. Take the torus-fixed point $x^{\epsilon}$ in $X_{r}$ corresponding to the maximal cone $\sigma$. Let $U$ be the torus-invariant open affine neighbourhood of $x^{\epsilon}$ in $X_{r}$. Then $U$ is an affine space of dimension $r$; in particular, $Pic(U)=0$. Therefore, we get

[TABLE]

and $Pic(X_{r})$ is generated by $\{D_{\rho_{i}^{\epsilon}}:1\leq i\leq r\}$ (see [Har70, Chapter II, Proposition 3.1, page 66]). Since $\{D_{\rho_{i}^{\epsilon}}:1\leq i\leq r\}$ is linearly independent and the rank of $Pic(X_{r})$ is $r$, this set $\{D_{\rho_{i}^{\epsilon}}:1\leq i\leq r\}$ forms a basis of $Pic(X_{r})$. ∎

By Lemma 3.1, the set $\{D_{\rho_{i}^{+}}:1\leq i\leq r\}$ forms a basis of $Pic(X_{r})$. Now we express for each $1\leq i\leq r$, $D_{\rho_{i}^{-}}$ in terms of $D_{\rho_{j}^{+}}$’s ($1\leq j\leq r$). Let $1\leq i\leq r$, define $h_{i}^{i-1}:=-\beta_{i(i-1)}$ and

[TABLE]

Then we prove,

Lemma 3.2**.**

Let $1\leq i\leq r$. The coefficient of $D_{\rho_{j}^{+}}$ in $D_{\rho_{i}^{-}}$ is $h_{i}^{j}$.

Proof.

Proof is by induction on $i$ and by using

[TABLE]

Recall the equation (2.2),

[TABLE]

If $i=1$, by (3.2), we see

[TABLE]

Then we have

[TABLE]

If $i=2$, by (3.2) and (2.2), we see

[TABLE]

By (3.3), we get

[TABLE]

By induction assume that

[TABLE]

Again by (3.2) and (2.2), we see

[TABLE]

Then

[TABLE]

Hence

[TABLE]

Since $h_{k}^{j}=0$ for $k<j$, we get

[TABLE]

Then

[TABLE]

Therefore, we conclude that $D_{\rho_{i}^{-}}\sim D_{\rho_{i}^{+}}+\sum_{j=1}^{i-1}h_{i}^{j}D_{\rho_{j}^{+}}$ . This completes the proof of the lemma. ∎

Let $\epsilon\in\{+,-\}$. Define $\Sigma(1)^{\epsilon}:=\{\rho_{i}^{\epsilon}:1\leq i\leq r\}.$ Then

[TABLE]

For $1\leq i\leq r$, let $g_{i}:=a_{\rho_{i}^{+}}+\sum_{j=i}^{r}a_{\rho_{j}^{-}}h_{j}^{i}.$ Then we have

Corollary 3.3**.**

$D=\sum_{\rho\in\Sigma(1)}a_{\rho}D_{\rho}\sim\sum_{i=1}^{r}g_{i}D_{\rho_{i}^{+}}.$**

Proof.

We have $D=\sum_{\rho\in\Sigma(1)}a_{\rho}D_{\rho}=\sum_{i=1}^{r}a_{\rho_{i}^{+}}D_{\rho_{i}^{+}}+\sum_{i=1}^{r}a_{\rho_{i}^{-}}D_{\rho_{i}^{-}}.$ By Lemma 3.2, we can see that $\sum_{i=1}^{r}a_{\rho_{i}^{-}}D_{\rho_{i}^{-}}\sim\sum_{i=1}^{r}a_{\rho_{i}^{-}}(\sum_{j=1}^{i}h_{i}^{j}D_{\rho_{j}^{+}}).$ Then $\sum_{i=1}^{r}a_{\rho_{i}^{-}}D_{\rho_{i}^{-}}\sim\sum_{i=1}^{r}(\sum_{j=i}^{r}a_{\rho_{j}^{-}}h_{j}^{i})D_{\rho_{j}^{+}}.$ Hence we have $D\sim\sum_{i=1}^{r}(a_{\rho_{i}^{+}}+\sum_{j=i}^{r}a_{\rho_{j}^{-}}h_{j}^{i})D_{\rho_{i}^{+}}.$ Thus, $D\sim\sum_{i=1}^{r}g_{i}D_{\rho_{i}^{+}}$ and this completes the proof. ∎

Remark 3.4**.**

By Corollary 3.3, we see some vanishing results of the cohomology of line bundles on BSDH varieties in [Cha17b].

Let $1\leq i\leq r$. We prove the following.

Lemma 3.5**.**

The relative tangent bundle $T_{\pi_{i}}$ of $\pi_{i}:X_{i}\to X_{i-1}$ is given by

[TABLE]

Proof.

By definition of Bott tower, $\pi_{i}$ is a $\mathbb{P}^{1}$-fibration. Then the relative canonical bundle $K_{\pi_{i}}$ is given by

[TABLE]

(see [Kle80, Corollary 24, page 56]). By [CLS11, Theorem 8.2.3] (see also [Ful93, Page 74]), we have

[TABLE]

Then

[TABLE]

where $\Sigma^{\prime}$ is the fan of $X_{i-1}$. Since $X_{i-1}$ smooth, any divisor of the form $D=\sum_{\rho^{\prime}\in\Sigma^{\prime}(1)}a_{\rho^{\prime}}D_{\rho^{\prime}}$ with $a_{\rho^{\prime}}\in\mathbb{Z}$, in $X_{i-1}$ is Cartier. Hence the pullback $\pi_{i}^{*}(D)$ is defined and given by

[TABLE]

where $\varphi_{D}$ is the support function corresponding to the divisor $D$ (see [CLS11, Theorem 4.2.12] for the correspondence between support functions and Cartier divisors). Since the lattice map $\overline{\pi}_{i}:\mathbb{Z}^{i}\to\mathbb{Z}^{i-1}$ is the projection onto the first $i-1$ factors (see page 6), by definition of $u_{\rho}$ and $e_{j}^{-}$ (see (2.2)), for $\epsilon\in\{+,-\}$ we have

[TABLE]

Hence

[TABLE]

Thus we have,

[TABLE]

Therefore, we see that

[TABLE]

By (2.2), we note that

[TABLE]

Since $\check{K}_{\pi_{i}}=det~{}T_{\pi_{i}}$, we get $\check{K}_{\pi_{i}}=T_{\pi_{i}}$ as $\pi_{i}$ is a $\mathbb{P}^{1}$-fibration. Therefore, the result follows from (3.4) and (3.5). ∎

Remark 3.6**.**

By Lemma 3.2, the relative tangent bundle $T_{\pi_{i}}$ can be expressed in terms of $D_{\rho_{i}^{+}}$ ($1\leq i\leq r$).

The following is well known and proved here for completeness.

Lemma 3.7**.**

Let $X$ and $Y$ be smooth varieties. Let $f:X\longrightarrow Y$ be a fibration with a section $\sigma$ and denote by $\sigma(Y)$ its image in $X$. Then the restriction of the relative tangent bundle $T_{f}$ to $\sigma(Y)$ is isomorphic to the normal bundle $\mathcal{N}_{\sigma(Y)/X}$ of $\sigma(Y)$ in $X$.

Proof.

Consider the normal bundle short exact sequence

[TABLE]

where $T_{\sigma(Y)}$ and $T_{X}$ are the tangent bundles of $\sigma(Y)$ and $X$ respectively. Also consider the following short exact sequence

[TABLE]

By restricting (3.7) to $\sigma(Y)$, since $\sigma$ is a section of $f$, we get the following short exact sequence

[TABLE]

By using (3.6) and (3.8), we see $T_{f}|_{\sigma(Y)}$ is isomorphic to $\mathcal{N}_{\sigma(Y)/X}$. This completes the proof. ∎

We prove,

Lemma 3.8**.**

Let $1\leq i\leq r$. The normal bundle $\mathcal{N}_{X_{i}/X_{i-1}}$ of $X_{i-1}$ in $X_{i}$ is $\check{\mathscr{L}_{i-1}}$, where $\mathscr{L}_{i-1}$ is as in the definition of Bott tower and $\check{\mathscr{L}_{i-1}}$ is denotes the dual of $\mathscr{L}_{i-1}$.

Proof.

Fix $1\leq i\leq r$ and let $\mathscr{L}:=\mathscr{L}_{i-1}$. Recall that $\mathbb{P}(\mathscr{E})$ is by definition $Proj(S(\mathscr{E}))$, $S(\mathscr{E})$ is symmetric algebra of $\mathscr{E}=\mathcal{O}_{X_{i-1}}\oplus\mathscr{L}$ (see [Har77, Page 162]). Let $V(\mathscr{L})=Spec(S(\mathscr{L}))$, the geometric vector bundle associated to the locally free sheaf (line bundle) $\mathscr{L}$ (see [Har77, Exercise 5.18, Page 128]). Then, $V(\mathscr{L})$ is an open subvariety in $\mathbb{P}(\mathscr{E})$ and we have the following commutative diagram

[TABLE]

Also note that the section $s^{0}_{i}(X_{i-1})$ of $\pi_{i}$ corresponding to the projection $\mathscr{E}\to\mathcal{O}_{X_{i}}$ is same as the zero section $\sigma_{\pi}(X_{i-1})$ of $\pi$. Now consider the following short exact sequence

[TABLE]

Since the restriction $T_{\pi|_{\sigma_{\pi}}(X_{i-1})}$ of $T_{\pi}$ to $\sigma_{\pi}(X_{r-1})$ is $\check{\mathscr{L}}$, by Lemma 3.7 and by above short exact sequence (3.9) we see that $\mathcal{N}_{\sigma_{\pi}(X_{i-1})/V(\mathscr{L})}\simeq\check{\mathscr{L}}.$ Hence we conclude that $\mathcal{N}_{X_{i-1}/X_{i}}\simeq\check{\mathscr{L}}$ (here we are identifying $X_{i-1}$ with the section corresponding to the projection $\mathscr{E}=\mathcal{O}_{X_{i-1}}\oplus\mathscr{L}\to\mathcal{O}_{X_{i-1}}).$ This completes the proof of the lemma. ∎

Let $1\leq i\leq r$. We prove,

Lemma 3.9**.**

1. (1)

The toric sections of $\pi_{i}$ are given by $D_{\rho_{i}^{\epsilon}},{\epsilon}\in\{+,-\}.$ 2. (2)

The normal bundle $\mathcal{N}_{X_{i-1}/X_{i}}$ of $X_{i-1}$ in $X_{i}$ is given by

[TABLE]

where the line bundle $\mathscr{L}_{i-1}$ is as in the definition of the Bott tower $X_{i}$.

Proof.

Proof of (1): Recall that $\pi_{i}$ is a $\mathbb{P}^{1}$-fibration induced by the projection $\overline{\pi}_{i}:\mathbb{Z}^{i}\to\mathbb{Z}^{i-1}$. For each cone $\sigma\in\Sigma_{F}$ of dimension 1 (which is a maximal cone in $\Sigma_{F}$, where $\Sigma_{F}$ denote the fan of the fiber $\mathbb{P}^{1}$), the subvariety $V(\sigma)$ is an invariant section of $\pi_{i}$, which is an invariant divisor in $X_{i}.$ Hence we get two invariant divisors $V(\rho_{i}^{+})=D_{\rho_{i}^{+}}$ and $V(\rho_{i}^{-})=D_{\rho_{i}^{-}}$.

Proof of (2): By Lemma 3.8, we have $\mathcal{N}_{X_{i-1}/X_{i}}=\check{\mathscr{L}_{i-1}}$ and the section $X_{i-1}$ is given by the projection $\mathscr{E}=\mathcal{O}_{X_{i-1}}\oplus\mathscr{L}_{i-1}\to\mathcal{O}_{X_{i-1}}$. Hence (2) follows from (1). ∎

4. Primitive relations of the Bott tower

4.1. Primitive collections and primitive relations

First recall the notion of primitive collections and primitive relations of a fan $\Sigma$, which are basic tools for the classification of Fano toric varieties due to Batyrev (see [Bat91]).

Definition 4.1**.**

We say $P\subset\Sigma(1)$ is a primitive collection if $P$ is not contained in $\sigma(1)$ for some $\sigma\in\Sigma$ but any proper subset is. Note that if $\Sigma$ is simplicial, primitive collection means that $P$ does not generate a cone in $\Sigma$ but every proper subset does.

Definition 4.2**.**

Let $P=\{\rho_{1},\ldots,\rho_{k}\}$ be a primitive collection in a complete simplicial fan $\Sigma$. Recall $u_{\rho}$ is the primitive vector of the ray $\rho\in\Sigma$. Then $\sum_{i=1}^{k}u_{\rho_{i}}$ is in the relative interior of a cone $\gamma_{P}$ in $\Sigma$ with a unique expression

[TABLE]

Then we call (4.1) the primitive relation of $X_{\Sigma}$ corresponding to $P.$

Recall that $TDiv(X_{\Sigma})$ denote the group of torus-invariant divisors in $X_{\Sigma}$ (see Page 4). Since the fan $\Sigma$ of $X_{r}$ is full dimensional, we have the following short exact sequence

[TABLE]

where the maps are given by $\varphi_{1}:m\mapsto div(\chi^{m})~{}~{}\mbox{and}~{}~{}\varphi_{2}:D\mapsto\mathcal{O}_{X_{r}}(D)$ (see [CLS11, Theorem 4.2.1]).

Now we recall some standard notations: Let $X$ be a smooth projective variety, we define

[TABLE]

where $\equiv$ is the numerical equivalence, i.e. $Z\equiv Z^{\prime}$ if and only if $D\cdot Z=D\cdot Z^{\prime}$ for all divisors $D$ in $X$. We denote by $[C]$ the class of $C$ in $N_{1}(X)_{\mathbb{Z}}$. Let $N_{1}(X):=N_{1}(X)_{\mathbb{Z}}\otimes\mathbb{R}$. It is a well known fact that $N_{1}(X)$ is a finite dimensional real vector space (see [Kle66, Proposition 4, §1, Chapter IV]). In the case where $X$ is a (smooth projective) toric variety, $N_{1}(X)_{\mathbb{Z}}$ is dual to $Pic(X)$ via the natural pairing (see [CLS11, Proposition 6.3.15]). In our case $X=X_{r}$, there are dual exact sequences:

[TABLE]

and

[TABLE]

where

[TABLE]

and

[TABLE]

(see [CLS11, Proposition 6.4.1]). Let $P$ be a primitive collection in $\Sigma$. Note that since $X_{r}$ is smooth projective, $P\cap\gamma_{P}(1)=\emptyset$ and

[TABLE]

(see [CLS11, Proposition 7.3.6]). As an element in $\mathbb{Z}^{\sum(1)}$, we write $r(P)=(r_{\rho})_{\rho\in\Sigma(1)}$, where

[TABLE]

Then by (4.1) we see that

[TABLE]

Hence by the exact sequence (4.3) and by (4.4), we observe that $r(P)$ gives an element in $N_{1}(X_{r})_{\mathbb{Z}}$ (see [CLS11, Page 305]). We prove,

Lemma 4.3**.**

Let $P_{i}:=\{\rho_{i}^{+},\rho_{i}^{-}\}$, $1\leq i\leq r$. Then $\{P_{i}:1\leq i\leq r\}$ is the set of all primitive collections of the fan $\Sigma$ of $X_{r}$.

Proof.

By Theorem 2.6, the cones in the fan $\Sigma$ of $X_{r}$ are generated by subsets of $\{e_{1}^{+},\ldots,e_{r}^{+},e_{1}^{-},\ldots,e_{r}^{-}\}$ and containing no subset of the form $\{e_{i}^{+},e_{i}^{-}\}$. Then by Definition 4.1, it is clear that $P_{i}=\{\rho_{i}^{+},\rho_{i}^{-}\}$ is a primitive collection for all $i$. Also note that again by description of the cones in $\Sigma$, any primitive collection must contain a $P_{i}$ for some $1\leq i\leq r$.

Fix $1\leq i\leq r$. Let $Q$ be a collection of one-dimensional cones such that it properly contains $P_{i}$, i.e. there exists $1\leq j\leq r$ and $j\neq i$ such that $\rho_{j}^{\epsilon}\in Q\supset P_{i}$, $\epsilon\in\{+,-\}$. Assume that $Q$ is a primitive collection. Then by Definition 4.1, $\{\rho_{i}^{+},\rho_{i}^{-}\}\subset Q$ generates a cone in $\Sigma$. This is a contradiction to the description of the cones in $\Sigma$. Therefore, we conclude that $\{P_{i}:1\leq i\leq r\}$ is the set of all primitive collections. ∎

Now we define the Contractible classes from [Cas03]: Let $X$ be a smooth projective toric variety. We define $NE(X)_{\mathbb{Z}}$ in $N_{1}(X)$ by

[TABLE]

Let $\gamma\in NE(X)_{\mathbb{Z}}$ be primitive (i.e. the generator of $\mathbb{Z}_{\geq 0}\gamma$) and such that there exists some irreducible curve in $X$ having numerical class in $\mathbb{Q}_{\geq 0}\gamma$. Then

Definition 4.4**.**

(see [Cas03, Definition 2.3]) The above class $\gamma$ is called contractible if there exists a toric variety $X_{\gamma}$ and an equivariant morphism $\phi_{\gamma}:X\to X_{\gamma}$, surjective with connected fibers, such that for every irreducible curve $C$ in $X$,

[TABLE]

Remark 4.5**.**

Note that any contractible class is always a class of some invariant curve and also a primitive relation (see [Cas03, Theorem 2.2] and [Sca09, Page 74]).

Recall the following result from [Cas03, Proposition 3.4].

Proposition 4.6**.**

Let $P=\{\rho_{1},\ldots,\rho_{k}\}$ be a primitive collection in $\Sigma$, with the primitive relation $r(P)$:

[TABLE]

Then $r(P)$ is contractible if and only if for every primitive collection $Q$ of $\Sigma$ such that $P\cap Q\neq\emptyset$ and $P\neq Q$, the set $(Q\setminus P)\cup\gamma_{P}(1)$ contains a primitive collection.

4.2. Mori cone

We use the notation as above. Let $X$ be a smooth projective variety. We define $NE(X)$ the real convex cone in $N_{1}(X)$ generated by classes of irreducible curves. The **Mori cone ** $\overline{NE}(X)$ is the closure of $NE(X)$ in $N_{1}(X)$ and it is a strongly convex cone of maximal dimension.

If $X$ is a (smooth projective) toric variety, it is known that $NE(X)_{\mathbb{Z}}$ is generated by the finitely many torus-invariant irreducible curves in $X$ and hence $NE(X)_{\mathbb{Z}}$ is a finitely generated monoid. Hence the cone $NE(X)=\overline{NE}(X)$ is a rational polyhedral cone and we have

[TABLE]

where $r=dim(X)$ and $[V(\tau)]\in N_{1}(X)_{\mathbb{Z}}$ is the class of the toric curve $V(\tau)$. This is called the Toric Cone Theorem (see [CLS11, Theorem 6.3.20]). Let $\tau\in\Sigma(r-1)$ be a wall, that is $\tau=\sigma\cap\sigma^{\prime}$ for some $\sigma,\sigma^{\prime}\in\Sigma(r)$. Let $\sigma$ (respectively, $\sigma^{\prime}$) is generated by $\{u_{\rho_{1}},u_{\rho_{2}},\ldots,u_{\rho_{r}}\}$ (respectively, by $\{u_{\rho_{2}},\ldots,u_{\rho_{r+1}}\}$) and let $\tau$ be generated by $\{u_{\rho_{2}},\ldots,u_{\rho_{r}}\}.$ Then we get a linear relation,

[TABLE]

The relation (4.6) called wall relation and we have

[TABLE]

(see [CLS11, Proposition 6.4.4 and eq. (6.4.6) page 303]). Now we describe the Mori cone $\overline{NE}(X_{r})$ of $X_{r}$ in terms of the primitive relations of $X_{r}$.

Theorem 4.7**.**

$\overline{NE}(X_{r})_{\mathbb{Z}}=\sum_{i=1}^{r}\mathbb{Z}_{\geq 0}r(P_{i})$.

Proof.

We have

[TABLE]

where $\mathscr{P}$ is the set of all primitive collections in $X_{r}$ (see [CLS11, Theorem 6.4.11]). By Lemma 4.3, $\{P_{i}:1\leq i\leq r\}$ is the set of all primitive collections of $X_{r}$. Therefore, we get

[TABLE]

By [Cas03, Theorem 4.1], we have

[TABLE]

where $\mathscr{C}$ is the set of all contractible classes in $X_{r}.$

By Proposition 4.6, we can see that the primitive relations $r(P_{i})$ are contractible classes for $1\leq i\leq r$. Since any contractible class is a primitive relation, we get

[TABLE]

Hence we conclude that

[TABLE]

This completes the proof of the theorem. ∎

We have

Corollary 4.8**.**

The set $\{r(P_{i}):1\leq i\leq r\}$ forms a basis of $N_{1}(X_{r})_{\mathbb{Z}}$.

Proof.

By Theorem 4.7, $\{r(P_{i}):1\leq i\leq r\}$ generates the monoid $\overline{NE}(X_{r})_{\mathbb{Z}}$ and the cone $\overline{NE}(X_{r})$ is of dimension $r$. So $r(P_{i})$ for $1\leq i\leq r$ are linearly independent. Also the group $N_{1}(X_{r})_{\mathbb{Z}}$ is generated by $\overline{NE}(X_{r})_{\mathbb{Z}}$, hence by $r(P_{i})$ for $1\leq i\leq r$. Hence these form a basis of $N_{1}(X_{r})_{\mathbb{Z}}$. ∎

Next we describe the primitive relation $r(P_{i})$ explicitly by finding the cone $\gamma_{P_{i}}$ in (4.1) for $1\leq i\leq r$. We also observe that these cones depend on the given matrix corresponding to the Bott tower. We need some notation to state the result. Recall the matrix $M_{r}$ corresponding to the Bott tower $X_{r}$ is

[TABLE]

(see Section 2). Fix $1\leq i\leq r$. Define:

1. (1)

Let $r\geq j>j_{1}=i\geq 1$ and define $a_{1,j}:=\beta_{j_{1}j}$. 2. (2)

Let $r\geq j_{2}>j_{1}$ be the least integer such that $a_{1,j}>0$, then define for $j>j_{2}$

[TABLE] 3. (3)

Let $k>2$ and let $r\geq j_{k}>j_{k-1}$ be the least integer such that $a_{k-1,j}<0$, then inductively, define for $j>j_{k}$

[TABLE] 4. (4)

For $j\leq i$, $b_{j}:=0$, and for $j>i$ define

[TABLE]

Note that we have

[TABLE] 5. (5)

Let $I_{i}:=\{j_{1},\ldots,j_{m}\}$.

Example 4.9**.**

Let

[TABLE]

Let $i=1$, then $j_{1}=1$ and (1) $a_{1,2}=\beta_{12}=-1$ ; (2) $a_{1,3}=\beta_{13}=-1$ ; (3) $a_{1,4}=\beta_{14}=-1$ ; (4) $a_{1,5}=\beta_{15}=2$ ; (5) $a_{1,6}=\beta_{16}=-1$ ; (6) $a_{1,7}=\beta_{17}=2$.

Then $j_{2}=5$ and (1) $a_{2,6}=\beta_{15}\beta_{56}-\beta_{16}=-1$ ; (2) $a_{2,7}=\beta_{15}\beta_{57}-\beta_{17}=2$ .

Then $j_{3}=6$ and $a_{3,7}=-a_{2,6}\beta_{67}+a_{2,7}=-(-1)(-1)+(2)=1$.

Therefore, $I_{1}=\{1,5,6\}$ .

Let $1\leq i\leq r$. Let $\mathscr{A}_{i}:=\{e_{j}^{\epsilon_{j}}:1\leq j\leq r,b_{j}\neq 0~{}\mbox{and}~{}$

[TABLE]

Remark 4.10**.**

Note that as $b_{j}=0$ for $j\leq i$, we can take $i<j\leq r$ in the definition of $\mathscr{A}_{i}$.

Now we have,

Proposition 4.11**.**

Let $1\leq i\leq r$. The cone $\gamma_{P_{i}}$ in the primitive relation of $X_{r}$ corresponding to $P_{i}$ is generated by $\mathscr{A}_{i}$.

Before going to the proof we see an example.

Example 4.12**.**

We use same setting as in Example 4.9. By Lemma 4.3, we have $P_{i}=\{\rho_{i}^{+},\rho_{i}^{-}\}$ for all $1\leq i\leq 7$. By definition of $e_{i}^{-}$(see (2.2)), we have

(i) $e_{1}^{-}+e_{1}^{+}=e_{2}^{+}+e_{3}^{+}+e_{4}^{+}-2e_{5}^{+}+e_{6}^{+}-2e_{7}^{+}$; (ii) $e_{2}^{-}+e_{2}^{+}=-2e_{4}^{+}+e_{5}^{+}-2e_{6}^{+}+e_{7}^{+}$; (iii) $e_{3}^{-}+e_{3}^{+}=e_{5}^{+}+e_{7}^{+}$; (iv) $e_{4}^{-}+e_{4}^{+}=e_{5}^{+}-2e_{6}^{+}+e_{7}^{+}$; (v) $e_{5}^{-}+e_{5}^{+}=e_{6}^{+}-2e_{7}^{+}$; (vi) $e_{6}^{-}+e_{6}^{+}=e_{7}^{+}$; (vii) $e_{7}^{-}+e_{7}^{+}=0.$

Now we describe the cone $\gamma_{P_{1}}$. Observe that in (i) coefficient of $e_{5}^{+}$ is negative. By (v), we can see

[TABLE]

Then $e_{1}^{-}+e_{1}^{+}=e_{2}^{+}+e_{3}^{+}+e_{4}^{+}+2e_{5}^{-}-e_{6}^{+}+2e_{7}^{+}.$ By (vi),

[TABLE]

In this case, $I_{1}=\{1,5,6\}$ (see Example 4.9) and the cone $\gamma_{P_{1}}$ is generated by

[TABLE]

Now we prove Proposition 4.11:

Proof.

By (2.2), for all $1\leq i\leq r$, we have

[TABLE]

If for all $j>i$, $\beta_{ij}\leq 0$, then the cone $\gamma_{P_{i}}$ is generated by $\{e^{+}_{j}:j>i,\beta_{ij}<0\}.$ If not, choose the least integer $j_{2}>i$ such that $\beta_{ij_{2}}>0$. Now write

[TABLE]

Again by using (4.9), we have

[TABLE]

Then

[TABLE]

By definition $a_{2,j}=\beta_{ij_{2}}\beta_{j_{2}j}-\beta_{ij}$, then we have

[TABLE]

If $a_{2,j}\geq 0$ for all $j>j_{2}$, then $\gamma_{P_{i}}$ is generated by

[TABLE]

Otherwise, choose the least integer $j_{3}>j_{2}$ such that $a_{2,j_{3}}<0$. By substituting $-e^{+}_{j_{3}}$ from (4.9), we get

[TABLE]

Then,

[TABLE]

By definition $a_{3,j}=-a_{2,j_{3}}\beta_{j_{3}j}+a_{2,j}$, then we have

[TABLE]

By repeating this process, we get the cone $\gamma_{P_{i}}$ as we required. ∎

Let $1\leq i\leq r$. Recall $I_{i}=\{i=j_{1},\ldots,j_{m}\}$ as in page 13. Define for $1\leq j\leq r$ ,

[TABLE]

Set $\gamma_{P_{i}}(1):=\{\gamma_{1},\ldots,\gamma_{l}\}$. Then we have

Corollary 4.13**.**

For $1\leq i\leq r$, the primitive relation $r(P_{i})(=(r_{\rho})_{\rho\in\Sigma(1)})$ of $X_{r}$ given by

[TABLE]

Example 4.14**.**

We use Example 4.12. The following can be seen easily from (4.8).

1. (1)

$\gamma_{P_{1}}(1)=\{\rho_{2}^{+},\rho_{3}^{+},\rho_{4}^{+},\rho_{5}^{-},\rho_{6}^{-},\rho_{7}^{+}\}$* .* 2. (2)

The primitive relation $r(P_{1})=(r_{\rho})_{\rho\in\Sigma(1)}$ is given by

[TABLE]

Now we describe the primitive relations $r(P_{i})$ in terms of intersection of two maximal cones in the fan of $X_{r}$. Let $1\leq i\leq r$. Let $\mathscr{C}^{\prime}_{i}:=\{e_{j}^{\epsilon_{j}}:1\leq j\leq r~{}\mbox{and}~{}$

[TABLE]

Let $\mathscr{C}^{\prime\prime}_{i}:=\{e_{j}^{\epsilon_{j}}:1\leq j\leq r~{}\mbox{and}~{}$

[TABLE]

Example 4.15**.**

We use Example 4.12, for $i=1$, we have $I_{1}=\{1,5,6\}$. Then

[TABLE]

We prove the following by using wall relation (see page 12).

Proposition 4.16**.**

Fix $1\leq i\leq r$. The class of curve $r(P_{i})$ is given by

[TABLE]

where $\tau_{i}=\sigma\cap\sigma^{\prime}$ and $\sigma$ (respectively, $\sigma^{\prime}$) is the cone generated by $\mathscr{C}^{\prime}_{i}$ (respectively, by $\mathscr{C}^{\prime\prime}_{i}$).

Proof.

From Corollary 4.13, we have the following.

[TABLE]

where $\epsilon_{j}$ is as in Proposition 4.11. First we show that the set $Q:=\{\rho\in\Sigma(1):D_{\rho}\cdot V(\tau_{i})>0\}$ is not contained in $\sigma(1)$ for any $\sigma\in\Sigma$ (we adapt the arguments of [CLS11, Proof of Theorem 6.4.11, page 306], here we are not assuming the curve $V(\tau_{i})$ is extremal). Indeed, suppose $Q\subseteq\sigma(1)$ for some $\sigma\in\Sigma$. Let $D$ be an ample divisor in $X_{r}$ (such exists as $X_{r}$ is projective). Then, we can assume that $D$ is of the form

[TABLE]

(see [CLS11, (6.4.10), page 306]). Then we can see

[TABLE]

As $Q\subseteq\sigma(1)$, by definition of $Q$, $D_{\rho}\cdot V(\tau_{i})\leq 0$ for $\rho\notin\sigma(1)$. Since $a_{\rho}\geq 0$ for $\rho\notin\sigma(1)$, we get $D\cdot V(\tau_{i})\leq 0$, which is a contradiction as $D$ is ample. Therefore, $Q$ is not contained in $\sigma(1)$ for any $\sigma\in\Sigma$. Hence to prove the proposition it is enough to prove

[TABLE]

(see again [CLS11, Proof of Theorem 6.4.11, page 306]). From (4.12) and by using wall relation, we can see that

[TABLE]

Since $c_{j}$’s are all positive integers (see (4.4)), by Lemma 4.3 we conclude that

[TABLE]

and hence $r(P_{i})=[V(\tau_{i})]$. This completes the proof of the proposition. ∎

Example 4.17**.**

In Example 4.12, the curve $r(P_{1})=[V(\tau_{1})]$ with $\tau_{1}=\sigma\cap\sigma^{\prime}$ where $\sigma$ is the cone generated by

[TABLE]

and $\sigma^{\prime}$ is the cone generated by

[TABLE]

Corollary 4.18**.**

$\overline{NE}(X_{r})_{\mathbb{Z}}=\sum_{i=1}^{r}\mathbb{Z}_{\geq 0}[V(\tau_{i})]$, where $\tau_{i}$ is as in Proposition 4.16.

Proof.

This follows from Theorem 4.7 and Proposition 4.16 ∎

5. Ample and nef line bundles on the Bott tower

Let $X$ be a smooth projective variety. Recall $N^{1}(X)$ is the real finite dimensional vector space of numerical classes of real divisors in $X$ (see [Kle66, §1, Chapter IV]). In $N^{1}(X)$, we define the nef cone $Nef(X)$ to be the cone generated by classes of numerically effective divisors and it is a strongly convex closed cone in $N^{1}(X)$. The ample cone $Amp(X)$ of $X$ is the cone in $N^{1}(X)$ generated by classes of ample divisors. Note that the ample cone $Amp(X)$ is interior of the nef cone $Nef(X)$ (see [Kle66, Theorem 1, §2, Chapter IV]). Recall that the nef cone $Nef(X)$ and the Mori cone $\overline{NE}(X)$ are closed convex cones and are dual to each other (see [Kle66, §2, Chapter IV] ) .

In our case, we have $Pic(X_{r})_{\mathbb{R}}=N^{1}(X_{r})$, as the numerical equivalence and linear equivalence coincide (see [CLS11, Proposition 6.3.15]).

In this section, we characterize the ampleness and numerically effectiveness of line bundles on $X_{r}$ and we study the generators of the nef cone of $X_{r}$. We use the notation as in Section 4. Let $D=\sum a_{\rho}D_{\rho}$ be a toric divisor in $X_{r}$ and for $1\leq i\leq r$, define

[TABLE]

Then we prove,

Lemma 5.1**.**

1. (1)

The divisor $D$ is ample if and only if $d_{i}>0$ for all $1\leq i\leq r$. 2. (2)

The divisor $D$ is numerically effective (nef) if and only if $d_{i}\geq 0$ for all $1\leq i\leq r$.

Proof.

Proof of (2): Recall that the primitive relation $r(P_{i})$ is given by

[TABLE]

(see page 11). First observe that we have the following

[TABLE]

(see [CLS11, Proposition 6.4.1, page 299]). Then by (4.5), we get

[TABLE]

By Lemma 4.3, we have $P_{i}=\{\rho_{i}^{+},\rho_{i}^{-}\}$. Then by Corollary 4.13, we get

[TABLE]

Since the nef cone $Nef(X_{r})$ and the Mori cone $\overline{NE}(X_{r})$ are dual to each other, the divisor $D$ is nef if and only if $D\cdot C\geq 0$ for all torus-invariant irreducible curves $C$ in $X_{r}$. By Theorem 4.7, we have

[TABLE]

Hence $D$ is nef if and only if $D\cdot r(P_{i})\geq 0$ for all $1\leq i\leq r$. Therefore, by (5.1), we conclude that the divisor $D$ is nef if and only if $d_{i}\geq 0$ for all $1\leq i\leq r.$ This completes the proof of (2).

Proof of (1): Recall that the divisor $D$ is ample if and only if its class in $Pic(X_{r})_{\mathbb{R}}$ lies in the interior of the nef cone $Nef(X_{r})$. Hence by using similar arguments as in the proof of $(2)$ and the toric Kleiman criterion for ampleness [CLS11, Theorem 6.3.13], we can see that $D$ is ample if and only if $d_{i}>0$ for all $1\leq i\leq r.$ ∎

Next we describe the generators of the nef cone $Nef(X_{r})$ of $X_{r}$.

Example 5.2**.**

Let $M_{2}=\begin{bmatrix}1&-1\\ 0&1\end{bmatrix}_{2\times 2}$. Then $X_{2}=\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(1))$, the Hirzebruch surface $\mathscr{H}_{1}$ and the rays $\rho_{1}^{+},\rho_{1}^{-},\rho_{2}^{+}$ and $\rho_{2}^{-}$ of the fan (shown below) of $X_{2}$ are generated by $e_{1}^{+},e^{-}_{1}=-e_{1}^{+}+e_{2}^{+},e_{2}^{+}$ and $-e_{2}^{+}$ respectively.

\rho_{1}^{+}$$\rho_{2}^{+}$$\rho_{1}^{-}$$\rho_{2}^{-}

*Figure. *** Fan of Hirzebruck surface $\mathscr{H}_{1}$.

The primitive relations $r(P_{1})$ and $r(P_{2})$ are given by

[TABLE]

By wall relation, we observe that

1. (1)

$D_{\rho_{1}^{+}}\cdot r(P_{1})=1$* and $D_{\rho_{1}^{+}}\cdot r(P_{2})=0$.* 2. (2)

$D_{\rho_{2}^{-}}\cdot r(P_{1})=0$* and $D_{\rho_{2}^{-}}\cdot r(P_{2})=1$.*

Then the dual basis of $\{r(P_{1})~{},~{}r(P_{2})\}$ is $\{D_{\rho_{1}^{+}}~{},~{}D_{\rho_{2}^{-}}\}.$ Hence the generators of the nef cone $Nef(\mathscr{H}_{1})$ are $D_{\rho_{1}^{+}}$ and $D_{\rho_{2}^{-}}$. Note that by Lemma 3.1, $Pic(\mathscr{H}_{1})$ is generated by $\{D_{\rho_{1}^{+}},D_{\rho_{2}^{-}}\}.$ Let $D=aD_{\rho_{1}^{+}}+bD_{\rho_{2}^{-}}\in Pic(\mathscr{H}_{1})$. Then

[TABLE]

(this gives back [CLS11, Example (6.1.16), page 273]).

Now we prove the similar results for $X_{r}$. For $1\leq m\leq r$, define

[TABLE]

Remark 5.3**.**

Note that the set $J_{m}$ is the collection of indices $i<m$ for which $u_{\rho_{m}^{+}}$ appear in the $\gamma_{P_{i}}$ part of the expression (4.1) for the primitive relation $r(P_{i}).$

We set $D_{1}:=D_{\rho_{1}^{+}}$, and for $m>1$ define inductively

[TABLE]

where $-c_{\rho_{m}^{+}}^{\gamma_{P_{k}}}$ is the coefficient of $e_{m}^{+}$ in the primitive relation $r(P_{k})$.

Example 5.4**.**

In Example 5.2, $D_{1}=D_{\rho_{1}^{+}}$, $J_{2}=\{1\}$ and $D_{2}=D_{1}+D_{\rho_{2}^{+}}$. By using (2.2), we see that $0\sim div(\chi^{e_{1}^{+}})\sim D_{\rho_{1}^{+}}-D_{\rho_{1}^{-}}$ and $0\sim div(\chi^{e_{2}^{+}})\sim D_{\rho_{2}^{+}}-D_{\rho_{2}^{-}}+D_{\rho_{1}^{-}}.$ Hence $D_{2}=D_{1}+D_{\rho_{2}^{+}}=D_{\rho_{2}^{-}}$.

Example 5.5**.**

In Example 4.12,

1. (1)

*Recall by (4.8), we have $e_{1}^{-}+e_{1}^{+}=e_{2}^{+}+e_{3}^{+}+e_{4}^{+}+2e_{5}^{-}+e_{6}^{-}+e_{7}^{+}.$ Then, *$\gamma_{P_{1}}(1)=\{\rho_{2}^{+},\rho_{3}^{+},\rho_{4}^{+},\rho_{5}^{-},\rho_{6}^{-},\rho_{7}^{+}\}.$ 2. (2)

$\gamma_{P_{2}}(1)=\{\rho_{4}^{-},\rho_{5}^{-},\rho_{6}^{+},\rho_{7}^{+}\}$*          (since $e_{2}^{+}+e_{2}^{-}=2e_{4}^{-}+e_{5}^{-}+e_{6}^{+}+e^{+}_{7}$) .* 3. (3)

$\gamma_{P_{3}}(1)=\{\rho_{5}^{+},\rho_{7}^{+}\}$*                  (since $e_{3}^{+}+e^{-}_{3}=e_{5}^{+}+e_{7}^{+})$ .* 4. (4)

$\gamma_{P_{4}}(1)=\{\rho_{5}^{+},\rho_{6}^{-},\rho_{7}^{-}\}$*                  (since $e_{4}^{+}+e_{4}^{-}=e^{+}_{5}+2e_{6}^{-}+e_{7}^{-}$ ). * 5. (5)

$\gamma_{P_{5}}(1)=\{\rho_{6}^{+},\rho_{7}^{-}\}$*                  (since $e_{5}^{+}+e_{5}^{-}=e_{6}^{+}+2e_{7}^{-}$ ).* 6. (6)

$\gamma_{P_{6}}(1)=\{\rho_{7}^{+}\}$*                  (since $e_{6}^{+}+e_{6}^{-}=e_{7}^{+}$ ). * 7. (7)

$\gamma_{P_{7}}(1)=\emptyset.$*                   (since $e_{7}^{+}+e_{7}^{-}=0$ ).*

Then ,

1. (1)

If $m=1$, then $D_{1}=D_{\rho_{1}^{+}}$. 2. (2)

If $m=2$, then $J_{2}=\{1\}$ and $c_{\rho_{2}^{+}}^{\gamma_{P_{1}}}=1$. Hence $D_{2}=D_{1}+D_{\rho_{2}^{+}}$. 3. (3)

If $m=3$, then $J_{3}=\{1\}$ and $c_{\rho_{3}^{+}}^{\gamma_{P_{1}}}=1$. Hence $D_{3}=D_{1}+D_{\rho_{3}^{+}}$. 4. (4)

If $m=4$, then $J_{4}=\{1\}$ and $c_{\rho_{4}^{+}}^{\gamma_{P_{1}}}=1$. Hence $D_{4}=D_{1}+D_{\rho_{4}^{+}}$. 5. (5)

If $m=5$, then $J_{5}=\{3,4\}$ and $c_{\rho_{5}^{+}}^{\gamma_{P_{3}}}=1$   ;   $c_{\rho_{5}^{+}}^{\gamma_{P_{4}}}=1$. Hence

[TABLE] 6. (6)

If $m=6$, then $J_{6}=\{2,5\}$ and $c_{\rho_{6}^{+}}^{\gamma_{P_{2}}}=1~{};~{}c_{\rho_{6}^{+}}^{\gamma_{P_{5}}}=1$. Hence

[TABLE] 7. (7)

If $m=7$, then $J_{7}=\{1,2,3,6\}$ and

[TABLE]

[TABLE]

We prove,

Proposition 5.6**.**

The set $\{D_{i}:1\leq i\leq r\}$ is dual basis of $\{r(P_{i}):1\leq i\leq r\}$.

Proof.

Fix $1\leq i\leq r$. By Proposition 4.16, the class of curve corresponding to the primitive relation $r(P_{i})$ is given by

[TABLE]

(where $\tau_{i}$ is described as in Proposition 4.16). From Corollary 4.13, the primitive relation $r(P_{i})(=[V(\tau_{i})])$ is

[TABLE]

where $\epsilon_{j}$ is as in Proposition 4.16. Note that this is the wall relation for the torus-invariant curve $V(\tau_{i})$. We prove

[TABLE]

By (5.2) and by wall relation, we have

[TABLE]

Hence by definition of $D_{m}$, it is clear that

[TABLE]

Now we claim $D_{m}\cdot V(\tau_{i})=0$ for all $m>i$. Assume that $m>i$ and write $m=i+j$, where $1\leq j\leq r-i.$ We prove the claim by induction on $j$. If $j=1$, then $D_{m}=D_{i+1}$.

Case 1: If $J_{i+1}=\emptyset$, then $D_{i+1}=D_{\rho_{i+1}^{+}}$. By (5.4), we see that $D_{i+1}\cdot V(\tau_{i})=0.$

Case 2: Assume that $J_{i+1}\neq\emptyset$.

Subcase 1: If $i\notin J_{i+1}$, then by (5.4) and (5.5), we can see that $D_{i+1}\cdot V(\tau_{i})=0.$

Subcase 2: If $i\in J_{i+1}$, then by (5.5), we have $D_{i+1}\cdot V(\tau_{i})=c_{\rho_{i+1}^{+}}^{\gamma_{P_{i}}}+(D_{\rho_{i+1}^{+}}\cdot V(\tau_{i})).$

By (5.4), $D_{\rho_{i+1}^{+}}\cdot V(\tau_{i})=-c_{\rho_{i+1}^{+}}^{\gamma_{P_{i}}}$ and hence $D_{i+1}\cdot V(\tau_{i})=0$. This proves the claim for $j=1$.

Now assume that $j>1$.

Case 1: If $J_{m}=\emptyset$, then by (5.4) and (5.5), we see that $D_{m}\cdot V(\tau_{i})=0.$

Case 2: Assume that $J_{m}\neq\emptyset$.

Subcase 1: If $i\notin J_{m}$, then by (5.4) and (5.5), we can see that

[TABLE]

By induction on $j$, $D_{k}\cdot V(\tau_{i})=0$ for all $i<k<m$. By (5.4), as $m>i$ and $m\notin J_{m}$ , we have $D_{\rho_{m}^{+}}\cdot V(\tau_{i})=0~{}.$ Hence we conclude that $D_{m}\cdot V(\tau_{i})=0.$ This completes the proof of the proposition. ∎

We have,

Theorem 5.7**.**

1. (1)

The nef cone $Nef(X_{r})$ of $X_{r}$ is generated by $\{D_{i}:1\leq i\leq r\}$. 2. (2)

The divisor $D=\sum_{i}a_{i}D_{i}$ is ample if and only if $a_{i}>0$ for all $1\leq i\leq r$.

Proof.

Since the nef cone $Nef(X_{r})$ is dual of the Mori cone $\overline{NE}(X_{r})$, (1) follows from Proposition 5.6.

Proof of (2): This follows from (1) as the ample cone $Amp(X_{r})$ is interior of the nef cone $Nef(X_{r})$. ∎

6. Fano and weak Fano Bott towers

In this section we describe the matrices $M_{r}$ such that the corresponding Bott tower $X_{r}$ is Fano or weak Fano. First recall the Iitaka dimension of a Cartier divisor $D$ in a normal projective variety $X$. Let

[TABLE]

where $\mathscr{L}(mD)~{}\mbox{is the line bundle associated to }mD.$ For $m\in N(D)$, we have a rational map

[TABLE]

If $N(D)$ is empty we define the Iitaka dimension $\kappa(D)$ of $D$ as $-\infty$. Otherwise we define

[TABLE]

Observe that $\kappa(D)\in\{-\infty,0,1,\ldots,dim(X)\}.$ We say $D$ is big if $\kappa(D)=dim(X)$ (see [Laz04, Section 2.2, page 139]). Note that an ample divisor is big .

Lemma 6.1**.**

Let $X$ be a smooth projective variety, let $U$ be an open affine subset of $X$. Let $D$ be an effective divisor with support $X\setminus U$. Then $D$ is big.

Proof.

It suffices to show that there exists an effective divisor $E$ with support $X\setminus U$ such that $E$ is big. Indeed, we then have $mD=E+F$ for some $m\geq 0$ and for some effective divisor $F$. Then $E+F$ is big and hence so is $D$.

There exists $f_{1},\ldots,f_{n}\in\mathcal{O}_{X}(U)$ algebraically independent over $\mathbb{C}$, where $n=dim(X)$. View $f_{1},\ldots,f_{n}$ as rational functions on $X$, then $f_{1},\ldots,f_{n}\in H^{0}(X,\mathcal{O}_{X}(E))$ for some effective divisor $E$ with support $X\setminus U$ (since $div(f_{i})$ is an effective divisor with support in $X\setminus U$ for $1\leq i\leq r$). Thus, the monomials in $f_{1},\ldots,f_{n}$ of any degree $m$ are linearly independent elements of $H^{0}(X,\mathcal{O}_{X}(mE))$. So $dim(H^{0}(X,\mathcal{O}_{X}(mE)))$ grows like $m^{n}$ as $m\to\infty$. Hence $E$ is big (see [Laz04, Corollary 2.1.38 and Lemma 2.2.3]) and this completes the proof. ∎

We get the following as a variant of Lemma 6.1.

Corollary 6.2**.**

Let $X$ be a smooth projective variety and $D$ be an effective divisor. Let $supp(D)$ denotes the support of $D$. If $X\setminus supp(D)$ is affine, then $D$ is big.

A smooth projective variety $X$ is called Fano (respectively, weak Fano) if its anti-canonical line bundle $-K_{X}$ is ample (respectively, nef and big). To describe our results we use the notation and terminology from Section 1 (see page 2). We prove,

Theorem 6.3**.**

1. (1)

$X_{r}$* is Fano if and only if it satisfies $I$.* 2. (2)

$X_{r}$* is weak Fano if and only if it satisfies $II$.*

Proof.

Proof of (2): We have

[TABLE]

(see [CLS11, Theorem 8.2.3] or [Ful93, Page 74]). The anti-canonical line bundle of any projective toric variety is big, since we have

[TABLE]

$(\mathbb{C}^{*})^{r}$ is an affine open subset of $X_{r}$, by Corollary 6.2, $-K_{X_{r}}$ is big.

By using Lemma 5.1, we prove that $-K_{X_{r}}$ is nef if and only if $X_{r}$ satisfies $II$.

Let $D=-K_{X_{r}}$. By (6.1) and by definition of $d_{i}$ for $D$ (see Lemma 5.1), we have

[TABLE]

Then by Lemma 5.1(2), $-K_{X_{r}}$ is nef if and only if $\sum_{\gamma_{j}\in\gamma_{P_{i}}(1)}c_{j}\leq 2$ for all $1\leq i\leq r$.

First assume that $-K_{X_{r}}$ is nef. Fix $1\leq i\leq r$. By above discussion, we have

[TABLE]

Since $c_{j}$’s are positive integers (see (4.4)), we get the following situation:

[TABLE]

Case 1: If $|\gamma_{P_{i}}(1)|=0$, then by definition of $\gamma_{P_{i}}$ (see Definition 4.2), we have

[TABLE]

Then $|\eta_{i}^{+}|=0=|\eta_{i}^{-}|$. Hence we see $X_{r}$ satisfies the condition $N_{i}^{1}$.

Case 2: If $|\gamma_{P_{i}}(1)|=1$, then there exists a unique $r\geq m>i,$ such that $\gamma_{m}\in\gamma_{P_{i}}(1)$ and the primitive relation is either

[TABLE]

or

[TABLE]

By (6.2), we get $c_{m}=1~{}\mbox{or}~{}2$.

Subcase (i): Assume that $c_{m}=1$. If the primitive relation is (6.3), then we can see that $|\eta_{i}^{+}|=0$ and $c_{m}=-\beta_{im}=1$. Then $\beta_{im}=-1$ and hence $X_{r}$ satisfies the condition $N_{i}^{1}$.

If the primitive relation is (6.4), then by comparing with (4.10), we get that $|\eta_{i}^{+}|>0$ and there exists $m>i$ such that $c_{m}=\beta_{im}=1$, $\beta_{ij}=0$ for all $j<m$ and $\beta_{im}\beta_{mj}-\beta_{ij}=0$ for $j>m$. Hence $X_{r}$ satisfies the condition $N_{i}^{2}$.

Subcase (ii): Assume that $c_{m}=2$. If the primitive relation $r(P_{i})$ is (6.3), then $|\eta_{i}^{+}|=0$ and $|\eta_{i}^{-}|=1$. So by (2.2), we have $2=c_{m}=-\beta_{im}$.

If the primitive relation $r(P_{i})$ is (6.4), then by comparing with 4.10, we get $|\eta_{i}^{+}|>0$ and there exists $m>i$ such that $c_{m}=\beta_{im}=2$, $\beta_{ij}=0$ for $j<m$ and $\beta_{im}\beta_{mj}-\beta_{ij}=0$ for $j>m$.

Hence $X_{r}$ satisfies the condition $N_{i}^{2}$.

Case 3: If $|\gamma_{P_{i}}(1)|=2$, then there exists $r\geq m_{2}>m_{1}>i$ with $\gamma_{m_{1}},\gamma_{m_{2}}\in\gamma_{P_{i}}(1)$ such that the primitive relation $r(P_{i})$ is

[TABLE]

Subcase (i): If the primitive relation is $r(P_{i})~{}:~{}e_{i}^{+}+e_{i}^{-}=c_{m_{1}}e_{m_{1}}^{+}+c_{m_{2}}e_{m_{2}}^{+}$, by (2.2) we see that $|\eta_{i}^{+}|=0$ and $|\eta_{i}^{-}|=2$ . By (6.2) and (4.4) ($c_{i}$’s are positive integers), we get

[TABLE]

Hence $X_{r}$ satisfies the condition $N_{i}^{2}.$

Subcase (ii): If the primitive relation is $r(P_{i})~{}:~{}e_{i}^{+}+e_{i}^{-}=c_{m_{1}}e_{m_{1}}^{+}+c_{m_{2}}e_{m_{2}}^{-}$, by comparing with (4.10) we see that $|\eta_{i}^{+}|>0$ and there exists $m_{2}>m_{1}>i$ such that $c_{m_{1}}=-\beta_{im_{1}}=1$, $c_{m_{2}}=\beta_{im_{2}}=1$, $\beta_{ij}=0$ for $m_{1}\neq j<m_{2}$ and $\beta_{im_{2}}\beta_{m_{2}j}-\beta_{ij}=0$ for $j>m_{2}$.

Subcase (iii): If the primitive relation is $r(P_{i})~{}:~{}e_{i}^{+}+e_{i}^{-}=c_{m_{1}}e_{m_{1}}^{-}+c_{m_{2}}e_{m_{2}}^{+}$, by (4.10) we see that $|\eta_{i}^{+}|>0$ and there exists $m_{2}>m_{1}>i$ such that $c_{m_{1}}=\beta_{im_{1}}=1$, $c_{m_{2}}=\beta_{im_{1}}\beta_{m_{1}m_{2}}-\beta_{im_{2}}=1$, $\beta_{ij}=0$ for $j<m_{1}$ and $\beta_{im_{1}}\beta_{m_{1}j}-\beta_{ij}=0$ for $m_{1}<j\neq m_{2}$.

Subcase (iv): If the primitive relation is $r(P_{i})~{}:~{}e_{i}^{+}+e_{i}^{-}=c_{m_{1}}e_{m_{1}}^{-}+c_{m_{2}}e_{m_{2}}^{-}$, by comparing with (4.11), we see that $|\eta_{i}^{+}|>0$ and there exists $m_{2}>m_{1}>i$ such that $c_{m_{1}}=\beta_{im_{1}}=1$, $1=c_{m_{2}}=-(\beta_{im_{1}}\beta_{m_{1}m_{2}}-\beta_{im_{2}})$, $\beta_{ij}=0$ for $j<m_{1}$, $\beta_{im_{1}}\beta_{m_{1}j}-\beta_{ij}=0$ for $m_{1}<j<m_{2}$ and $\beta_{m_{2}j}+(\beta_{im_{1}}\beta_{m_{1}j}-\beta_{ij})=0$ for $j>m_{2}$. Hence $X_{r}$ satisfies the condition $N_{i}^{2}$.

Therefore, we conclude that if $X_{r}$ is weak Fano then $X_{r}$ satisfies the condition $II$. Similarly, we can prove by using Lemma 5.1(2), if $X_{r}$ satisfies $II$ then $X_{r}$ is weak Fano. This completes the proof of (2).

Proof of (1): This follows by using similar arguments as in the proof of (2) and Lemma 5.1(1). ∎

Remark 6.4**.**

In [YS18], S. Yusuke classified Fano (and weak Fano) “generalized Bott Manifolds”.

6.1. Local rigidity of Bott towers

Now we prove some vanishing results for the cohomology of tangent bundle of the Bott tower $X_{r}$ and we get some local rigidity results. Let $T_{X_{r}}$ denotes the tangent bundle of $X_{r}$. Then we have

Corollary 6.5**.**

If $X_{r}$ satisfies $I$, then $H^{i}(X_{r},T_{X_{r}})=0$ for all $i\geq 1$.

Proof.

If $X_{r}$ satisfies $I$, then by Theorem 6.3, $X_{r}$ is Fano variety. By [BB96, Proposition 4.2], since $X_{r}$ is a smooth Fano toric variety, we get $H^{i}(X_{r},T_{X_{r}})=0~{}\mbox{for all}~{}i\geq 1.$ ∎

It is well known that by Kodaira-Spencer theory, the vanishing of $H^{1}(X,T_{X})$ implies that $X$ is locally rigid, i.e. admits no local deformations (see [Huy06, Proposition 6.2.10, page 272]). Then by above result we have

Corollary 6.6**.**

The Bott tower $X_{r}$ is locally rigid if it satisfies $I$.

7. Log Fanoness of Bott towers

Recall that a pair $(X,D)$ of a normal projective variety $X$ and an effective $\mathbb{Q}$-divisor $D$ is Kawamata log terminal (klt) if $K_{X}+D$ is $\mathbb{Q}$-Cartier, and for all proper birational maps $f:Y\longrightarrow X$, the pull back $f^{*}(K_{X}+D)=K_{Y}+D^{\prime}$ satisfies $f_{*}K_{Y}=K_{X}$ and $\lfloor D^{\prime}\rfloor\leq 0$, where $\lfloor~{}\sum_{i}a_{i}D_{i}\rfloor=\sum_{i}\lfloor a_{i}\rfloor D_{i}$, $\lfloor x\rfloor$ is the greatest integer $\leq x$. The pair $(X,D)$ is called log Fano if it is klt and $-(K_{X}+D)$ is ample.

We recall here, a condition for the anti-canonical line bundle to be big (see [CG13]). Let $X$ be a $\mathbb{Q}$- Gorenstein projective normal variety over $\mathbb{C}$. If $X$ admits a divisor $D$ with the pair $(X,D)$ being log Fano then $-K_{X}$ is big (In [CG13] there is a necessary and sufficient condition that $X$ is log Fano (or “Fano type ”) variety, see [CG13, Theorem 1.1] for more details on this ).

If $X$ is smooth and $D$ is a normal crossing divisor, the pair $(X,D)$ is log Fano if and only if $\lfloor D\rfloor=0$ and $-(K_{X}+D)$ is ample (see [KM08, Lemma 2.30, Corollary 2.31 and Definition 2.34]). In case of toric variety $X$ see also [CLS11, Definition 11.4.23 and Proposition 11.4.24, page 558]. We use notation as in Lemma 5.1. Let $D=\sum_{\rho\in\Sigma(1)}a_{\rho}D_{\rho}$ be a toric divisor in $X_{r}$, with $a_{\rho}^{\prime}s$ in $\mathbb{Q}_{\geq 0}$ and $\lfloor D\rfloor=0$. For $1\leq i\leq r$, define

[TABLE]

Then we prove,

Theorem 7.1**.**

The pair $(X_{r},D)$ is log Fano if and only if $k_{i}<0$ for all $1\leq i\leq r$.

Proof.

From the above discussion by the condition on $D$, the pair $(X_{r},D)$ is log Fano if and only if $-(K_{X_{r}}+D)$ is ample. Note that as $-K_{X_{r}}=\sum_{\rho\in\Sigma(1)}D_{\rho}$, we get

[TABLE]

By Lemma 5.1, $-(K_{X_{r}}+D)$ is ample if and only if

[TABLE]

Recall the definition of $d_{i}$ for $D$,

[TABLE]

Then we have

[TABLE]

Hence in (7.1)

[TABLE]

and we conclude that $-(K_{X_{r}}+D)$ is ample if and only if $k_{i}<0$ for all $1\leq i\leq r$. This completes the proof of the theorem. ∎

8. Extremal rays and Mori rays of the Bott tower

In this section we study the extremal rays and Mori rays of Mori cone of $X_{r}$. First we recall some definitions. Let $V$ be a finite dimensional vector space over $\mathbb{R}$ and let $K$ be a (closed) cone in $V$. A subcone $Q$ in $K$ is called extremal if $u,v\in K,u+v\in Q$ then $u,v\in Q$. A face of $K$ is an extremal subcone. A one-dimensional face is called an extremal ray. Note that an extremal ray is contained in the boundary of $K$.

Let $X$ be a smooth projective variety. An extremal ray $R$ in $\overline{NE}(X)\subset N_{1}(X)$ is called Mori if $R\cdot K_{X}<0$, where $K_{X}$ is the canonical divisor in $X$. Recall that $\overline{NE}(X_{r})$ is a strongly convex rational polyhedral cone of maximal dimension in $N_{1}(X_{r})$. We prove,

Theorem 8.1**.**

1. (1)

The class of curves $r(P_{i})$ for $1\leq i\leq r$ are all extremal rays in the Mori cone $\overline{NE}(X_{r})$ of $X_{r}$. 2. (2)

Fix $1\leq i\leq r$, the class of curve $r(P_{i})$ is Mori ray if and only if either $|\gamma_{P_{i}}(1)|=0$, or $|\gamma_{P_{i}}(1)|=1$ with $c_{j}=1$ for $\gamma_{j}\in\gamma_{P_{i}}(1)$.

Proof.

Proof of (1): This follows from Theorem 4.7 and Corollary 4.8.

Proof of (2): By (1), $r(P_{i})$ $1\leq i\leq r$ are all extremal rays in $\overline{NE}(X_{r})$. Hence for $1\leq i\leq r$, $r(P_{i})$ is Mori if $K_{X{r}}\cdot r(P_{i})<0$. Since $K_{X_{r}}=-\sum_{\rho\in\Sigma(1)}D_{\rho}$, we can see by Corollary 4.13 and by similar arguments as in the proof of Lemma 5.1,

[TABLE]

Thus if $K_{X{r}}\cdot r(P_{i})<0$, then

[TABLE]

As $c_{j}$ are all positive integers (see (4.4)), we get either $|\gamma_{P_{i}}(1)|=0$, or $|\gamma_{P_{i}}(1)|=1$ and $c_{j}=1$ for $\gamma_{j}\in\gamma_{P_{i}}(1)$. Similarly, by using (8.1) we can prove the converse. This completes the proof of the theorem. ∎

Acknowledgements: I would like to thank Michel Brion for valuable discussions, many critical comments and for encouragement throughout the preparation of this article.