come

Experimental study of the combined effect of temperature and mixture parameters on the premature cracking of cement-based materials

Arnaud Pertue, Pierre Mounanga, Abdelhafid Khelidj, Denis Fournol

INTRODUCTION

At an early age, the volume changes of cementbased materials are the combined result of the physico-chemical evolution, the hydration heat release and the drying of the material. An inaccurate estimation of the amplitude and the rate of these volume variations can constitute an important cause of concrete durability problems.

In low water-to-cement ratio concretes (W/C < 0.5), the high cement proportioning results in an increase in the mechanical strength but also induces an augmentation of both autogenous and thermal deformations, in particular at a very early age. These deformations generate, when restrained, microscopic cracking (around the aggregates) or crossing cracks.

The knowledge of the deformation evolution in free conditions is not sufficient to precisely predict the risk of early-age cracking. Indeed, in restrained conditions, the relaxation phenomenon of the hydrating matrix attenuates the amplitude of autogenous shrinkage: the strain evolution is then different from that recorded in free conditions. It is therefore necessary to develop specific devices enabling the quantification of deformation in restrained conditions, taking into account the thermal history of the material. The parameters conditioning the development of early-age deformations are numerous: the cement type, the concrete's composition, the geometry and dimensions of the specimen, the curing temperature, the drying conditions, etc. The effects of some of these parameters have been investigated in previous studies [START_REF] Grzybowski | Shrinkage cracking of fiber reinforced concrete[END_REF][START_REF] Weiss | Prediction of early age shrinkage cracking in concrete[END_REF][START_REF] Weiss | Restrained shrinkage testing: the impact of specimen geometry on quality control testing for material performance assessment[END_REF][START_REF] Hossain | The role of specimen geometry and boundary conditions on stress development and cracking in the restrained ring test[END_REF]), but early-age autogenous cracking and the temperature effect on this type of cracking did not receive much attention.

Several testing techniques exist for the determination of restrained shrinkage and cracking risk of cement-based materials [START_REF] Bentur | Evaluation of early age cracking characteristics in cementitious systems[END_REF]. Among them, an easy and simple method is the ring test. It consists in casting a specimen of cementbased material around a metal ring and in measuring the dimensional variations of this ring. The restrained shrinkage causes the development of tensile stresses in the material. As soon as they become higher than the material tensile strength, a transversal crack appears in the specimen.

The ring test method has been used in many research works (Table 1) for the quantification of the cracking risk of cement-based materials. A bibliography and an analysis of these studies have been recently presented by [START_REF] Radlinska | The ring test: a review of recent developments[END_REF].

The present work focuses on the early-age autogenous cracking in cement paste. The effects of three parameters are measured: the isothermal curing temperature, the degree of restraint and the initial amount of water (W/C ratio). The ring test device used in this study is equipped with a peripheral hermetic thermal regulation system, which enables to ensure quasi-isothermal and autogenous conditions during the test.

Experimental study on the combined effect of temperature and mix parameters on the premature cracking of cement-based materials A. Pertué & P. Mounanga & A. Khelidj & D. Fournol GeM,UMR CNRS 6183,Research Institute in Civil Engineering and Mechanics,IUT St Nazaire,France ABSTRACT: Cracking can occur in early-age concrete when thermal, hydrous or physico-chemical deformations are restrained. In addition to a deterioration of the aesthetic aspect, this premature cracking generates a loss of durability of the structure. In this article, the autogenous cracking of cement-based materials is studied using the ring test method. A specific experimental system based on this method was developed. It enables to keep the specimens in autogenous and temperature-controlled conditions during the whole test duration. The influence of both temperatures between 20°C and 40°C and the degree of restraint on the first autogenous cracking age of CEM I cement pastes with water-to-cement ratio W/C = 0.3 and 0.4 has been examined. Increasing the curing temperature and the degree of restraint and decreasing the W/C ratio leads to a shortening of the cracking age. Moreover, the level of maturity (hydration degree) corresponding to the age of the first autogenous cracking decreases when the curing temperature increases. Two water-to-cement ratios (W/C = 0.3 and 0.4) and three curing temperatures (T = 20°C, 30°C and 40°C) are studied.

Quasi-isothermal ring-test device

The ring test device is schematized in Figures 1a and1b. Figure 2 presents an overall photograph of the experimental system.

Each device is composed of two concentric rings: a PVC external ring and a metallic central ring. Three different metals (brass, steel and stainless steel) are used. The characteristics of these central rings are given in Table 2. Each cement paste specimen cast between the two rings has a cross section of 40 × 40 mm.

The strain of the central annulus due to the cement paste shrinkage is measured by four strain gauges (Fig. 1b). Four additional gauges are placed inside the central ring for thermal compensation.

In order to carry out the tests at a controlled temperature, each device is equipped with a peripheral thermal regulation system made of a copper tube network joined on two aluminium plates and fed by a thermostated water bath (± 0.1°C) via a pump (Fig. 2). To avoid thermal losses, insulating foam is placed on the copper tubes and each device is set in a wooden box (Fig. 1a).

Testing protocol

The water circulation in the thermal regulation system is activated two hours before the casting of cement paste in order to stabilize the temperature of the rings. Form oil is applied on the rings and on the bottom of the device to limit friction between the specimen and the ring walls.

Each cement paste is prepared by mixing cement with water for 3 minutes. The specimen is then immediately cast around the central metal ring. A thermocouple, embedded in the sample, makes it possible to follow the temperature of the material during the test. The wooden box is then sealed and the automatic measurement acquisition begins. The time lag between the end of mixing and the first strain measurement does not exceed 15 minutes. The deformation of the central ring and the temperature of the specimen are recorded by a data logging system every 20 minutes. The test is stopped at the appearance of the first crossing crack (Fig. 3) detected by a brutal jump on the strain curve of the metal ring.

TEST RESULTS

Validation tests

Temperature control

The hydration heat can cause a significant elevation of the cement paste temperature during the first hours after mixing. It is essential to control these temperature variations in order to eliminate potential problems of thermal deformations.

Figure 4 shows the typical temperature curve of specimens during the tests. The beginning of each curve is marked by a temperature variation of about ± 2°C explained by the strong hydration heat release and the thermal inertia of the regulation system. This thermal instability lasts approximately 6 hours. Beyond this period, a maximal thermal deviation of 0.5°C is observed. The efficiency of the thermal regulation system can thus be considered as correct. The repeatability level of the test method is acceptable. The strain curves follow the same evolution. The absolute difference in maximal strain amplitude and in cracking age is about 2 μm/m and 2.3 hours, respectively.

Parametric study

In the three next paragraphs, the effects of the ring nature, the water-to-cement ratio and the temperature on the appearance of the first crack will be analysed.

First crossing crack

Effect of the degree of restraint

Recently, [START_REF] Hossain | The role of specimen geometry and boundary conditions on stress development and cracking in the restrained ring test[END_REF] studied the effect of the degree of restraint by measuring the influence of the ring thickness on the age of the first crack in drying conditions. They noted that the increase of the degree of restraint causes a shortening of the cracking age.

In the present work, the effect of the degree of restraint is investigated by using metal rings with different elastic moduli (brass, stainless steel and steel). The degree of restraint R is computed using the expression proposed by [START_REF] See | Shrinkage cracking characteristics of concrete using ring specimens[END_REF]:

c E c A m E m A m E m A R ⋅ + ⋅ ⋅ = (1)
where A c and A m are the section areas of the cement paste specimen (A c = 16 cm²) and the metal ring (A m = 4 cm²), respectively. E c and E m are the elastic moduli of the cement paste and the metal, respectively.

In the case of cement paste with W/C = 0.3, the values of the degree of restraint computed are 64% for the steel ring, 63% for the stainless steel ring and 46% for the brass ring. The effect of this degree of restraint can be seen on figure 6, which provides the results of the ring tests performed with the cement paste specimens prepared with W/C = 0.3 and cured at 30°C. From the ring strain ε m , it is possible to calculate the residual stress σ residual at the interface between the metal ring and the cement paste as [START_REF] Weiss | Restrained shrinkage cracking: the role of shrinkage reducing admixtures and specimen geometry[END_REF][START_REF] Hossain | Assessing residual stress development and stress relaxation in restrained concrete ring specimens[END_REF]):

R C m E t m t residual . . ) ( ) ( ε σ - = (2)
where C R is the ring geometry coefficient:

2 2 2 2 2 2 2 2 OS R IS R OS R OS R OC R OC R OS R R C - × - + =
(3) with R OS = outer metal radius; R OC = outer cement paste ring radius; and R IS = inner metal radius.

Figure 7 shows the evolution of the residual stress as a function of time for the three metal rings.

The three curves plotted in Figure 6 (and in Fig. 7) experience the same type of evolution: firstly a period of about 24 hours with low ring deformations (residual stress), then a faster augmentation of the strains (of the residual stress) and finally a brutal jump (drop) corresponding to the cracking of the specimens.

Figures 6 and7 demonstrate that the nature of the central ring influences both the age of cracking and the maximal ring strain measured. The test performed with the brass ring, which has the lower rigidity, shows the later the age of cracking, the higher the ring strain and the residual stress: a reduction of 13% of the degree of restraint leads to double the amplitude of the ring strain and to increase the maximal residual stress of about 15%.

A difference of about 10h is recorded between the age of cracking obtained with the steel ring and that measured with the stainless steel. This difference cannot be explained only by the slight difference existing between the elastic moduli of these two rings. The surface quality of the rings, which induces more or less friction at the interface with the cement paste specimen, could also play an important role in this cracking age difference. The metal surface in contact with the specimens was observed with a videomicroscope for each ring (Fig. 8). Different scratch thicknesses have been observed for the three rings. It can thus be supposed that the intensity of friction forces will not be the same for the three types of rings. However at the current state of our knowledge, it is still difficult to quantify the effect of the surface quality on the difference in cracking age. Tests are in progress to determine precisely the depth of roughness of each ring.

Brass

Steel Stainless steel

Curing temperature effect

The second parameter studied is the curing temperature. Figure 9 gives the ring test results obtained for the specimens with W/C = 0.4 at 20, 30 and 40°C. A preliminary point, which has to be underlined, is that, in the range considered, the temperature did not significantly modify the elastic modulus of metals: for instance, for stainless steel, it is 200 GPa at 20°C and 199 GPa at 40°C. In the following, the influence of this variation will be thus neglected.

Figure 9 shows that a higher curing temperature causes an acceleration of the early-age cracking: from 20 to 40°C, the cracking age is divided by 8.8. This phenomenon can be attributed to the thermoactivation of the cement hydration process at a very early age: when the temperature increases, the deformations of the cement paste develop faster [START_REF] Turcry | Can the maturity concept be used to separate the autogenous shrinkage and thermal deformation of a cement paste at early age?[END_REF][START_REF] Mounanga | Autogenous deformations of cement pastes -Part I: temperature effects at early age and micro-macro correlations[END_REF]) and the value of cracking tensile stress is reached more quickly.

In fact, this acceleration is the result of the competition between several complex phenomena: on one hand, the increase of internal tensile stress due to autogenous shrinkage (partially counterbalanced by the creep relaxation of the hydrating matrix) and, on the other hand, the development of the material tensile strength. When the internal stress becomes higher than the tensile strength, the specimen cracks (Fig. 10). Temperature accelerates both the internal stress rate and the tensile strength evolution. According to our results, the fact that early-age cracking occurs sooner when the curing temperature is higher seems to demonstrate that the thermoactivation of the internal stress rate is more important than the thermo-activation of the tensile strength evolution.

Besides, a second effect of the increase of curing temperature is the diminution of the maximal ring strain measured just before the cracking of the specimen. Between 20°C and 30°C and for all the specimens investigated, a significant diminution of the maximal ring strain is recorded when temperature increases whereas the difference between the amplitudes of the maximal ring strain obtained at 30°C and at 40°C is much lower (Fig. 9). Autogenous shrinkage is the driving-mechanism leading to early-age cracking, whereas both the rigidity evolution and the tensile strength development of the material are opposed to the increase of these deformations. Previous studies have already shown that a lower water/cement ratio induces an acceleration of the evolution of autogenous deformations (e.g. [START_REF] Baroghel-Bouny | Effect of mixparameters on autogenous deformations of cement pastes -Microstructural interpretations[END_REF], tensile strength and elastic modulus. The ring strain curves provided in Figures 11 and12 show that, among the main phenomena involved (autogenous deformation rate, evolution of rigidity and tensile strength development), the preponderant one remains the autogenous strain rate, since the gain of both rigidity and tensile strength due to the diminution of the initial water amount did not enable to slow down the occurrence of early-age cracking.

AUTOGENOUS EARLY-AGE CRACKING AND THE HYDRATION DEGREE CONCEPT

The test results discussed in the previous sections are expressed as a function of time. This mode of presentation is not optimal when comparing materials of different compositions since it does not enable to directly relate the cracking age to the physicochemical evolution of the material.

In the following, the ring test results are plotted as a function of the maturity level of the cement pastes expressed in terms of the hydration degree.

The evolution of the cement hydration degree has been computed with a program developed by the NIST, CEMHYD3D [START_REF] Bentz | CEMHYD3D: A three-dimensional cement hydration and microstructure development modelling package Version 3[END_REF].

The input data are the particle size distribution, the chemical composition and the apparent activation energy of cement. For this latter parameter, the value considered is 40 kJ/mol [START_REF] Mounanga | Autogenous deformations of cement pastes -Part I: temperature effects at early age and micro-macro correlations[END_REF].

Figures 13 and14 show the influence of the W/C ratio and the curing temperature on the hydration degree of the cement pastes. In the range of the W/C ratio investigated, the effect of this parameter remains weak at a very early age. On the contrary, a significant acceleration of the hydration rate is observed when the temperature increases, in accordance with the thermo-activation principle.

Figures 15 to 18 present the evolution of the ring strain as a function of the cement hydration degree for different cement pastes.

It can be noted that the augmentation of the curing temperature results in a decrease of the maturity level corresponding to the first autogenous cracking time. These results show that, when considering a relatively large temperature range (20 to 40°C), the thermal effect on the first early-age autogenous cracking is not only a kinetic one : temperature influences both the age of cracking and the maturity level corresponding to this cracking age.

CONCLUSIONS

From the experimental results presented in this paper, the following points can be underlined, concerning:

-The curing temperature. A higher curing temperature causes an acceleration of the early age autogenous cracking. This phenomenon can be attributed to the thermo-activation of hydration, which induces an increase of the autogenous shrinkage rate and therefore of the internal tensile stress. Temperature also influences the maximal amplitude of ring strain recorded just before the cracking. -The degree of restraint. It was observed that the higher the degree of restraint the sooner the autogenous cracking age. Besides, it seems that the surface quality of the ring has an impact on the early age cracking. Supplementary tests are in progress in order to better quantify the effect of this parameter. -The water/cement ratio. A diminution of the initial water amount leads to faster cracking of the cement pastes. This effect has been explained by the fact that, in low water-to-ratio cement-based systems, the autogenous shrinkage rate, responsible for the development of internal tensile stress, is higher.

Finally, from computation results obtained with CEMHYD3D, it was shown that, for a given value of the W/C ratio and in the temperature range studied, the level of maturity (hydration degree) corresponding to the autogenous cracking age decreases when the curing temperature increases.

Figure 1 .

 1 Figure 1. Diagrams of the quasi-isothermal ring-test device: aside view and b-top view.

Figure 2 .

 2 Figure 2. Photography of the experimental system.

Figure 3 .

 3 Figure 3. Transversal cracking in the cement paste annulus.

Figure 4 .

 4 Figure 4. Temperature control of the cement paste specimens.

Figure 5 .

 5 Figure 5. Repeatability of the ring tests (Brass ring, W/C = 0.3, T = 40°C).

  Figure 6. Ring strain for cement paste with W/C = 0.3 at T = 30°C.

Figure 7 .

 7 Figure 7. Residual stress for cement paste with W/C = 0.3 at T = 30°C.

Figure 9 .

 9 Figure 9. Brass ring strain for cement paste with W/C = 0.4 at T = 20, 30 and 40°C.

Figure 10 .Figure 11 .Figure 12 .

 101112 Figure 10. Competition between internal stress evolution and strength development in hydrating cementitious systems (Weiss 1999).

Figure 13 .Figure 14 .Figure 15 .Figure 16 .Figure 17 .Figure 18 .

 131415161718 Figure 13. CEMHYD3D hydration degree of cement pastes with W/C = 0.3 and W/C = 0.4 at T = 20°C.

Table 1 .

 1 Recent studies using the ring test method.

	_________________________________________________
	References	Mixture parameters Test conditions _______________ ____________
	C _________________________________________________ W/C A/C T (°C) RH (%)
	Grzybowski &
	Shah 1990	1	0.5	2	20	40
	Weiss &
	Ferguson 2001	1	0.5	1.5	21	50
	Weiss &
	Shah 2002	NSC 0.5	2.8	30	40
	Hossain &
	Weiss 2006	1	0.3-0.4 1.5-1.4 23	50
	Shah &
	Weiss 2006 _________________________________________________ 1 0.5 1.8 23 50
	C: cement type; NSC: normal strength concrete; W/C: water-
	to-cement ratio; A/C: aggregate-to-cement ratio; T: curing tem-
	perature; RH: relative humidity
	2 EXPERIMENTAL PROGRAM
	2.1 Materials
	French CEM I cement is used for all the experi-
	ments. According to Bogue's formula, it is com-
	posed of 60.4% of C 3 S, 12.2% of C 2 S, 8.8% of C 3 A,
	8.0% of C 4 AF and 7.4% of gypsum; its specific sur-
	face is 3390 cm²/g.

Table 2 .

 2 Characteristics of the metal rings. : inner metal radius; R OS : outer metal radius; E m : elastic modulus; υ: Poisson coefficient.

	___________________________________________ Designation R IS R OS E m υ
	mm ___________________________________________ mm MPa -
	Brass	95	105	100 000 0.33
	Steel	95	105	211 000 0.30
	Stainless Steel ___________________________________________ 95 105 200 000 0.30
	R

IS


B Narasimha Chary

ON MORI CONE OF BOTT TOWERS

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

A Bott tower of height r is a sequence of projective bundles

where

) for a line bundle L i-1 over X i-1 for all 1 ≤ i ≤ r and 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.

Introduction

In [START_REF] Bott | Applications of the theory of Morse to symmetric spaces[END_REF], R. Bott and H. Samelson introduced a family of (smooth differentiable) manifolds which may be viewed as the total spaces of iterated P 1 -bundles over a point {pt}, where each P 1 -bundle is the projectivization of a rank 2 decomposable vector bundle. In [START_REF] Grossberg | Bott towers, complete integrability, and the extended character of representations[END_REF], 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 [START_REF] Civan | Bott towers, crosspolytopes and torus actions[END_REF], Y. Civan proved that these are smooth projective toric varieties. These are called Bott towers, we denote them by {(X i , π i ) : 1 ≤ i ≤ r}, where

X r πr -→ X r-1 π r-1 -→ • • • π 2 -→ X 1 = P 1 π 1 -→ {pt}, X i = P(O X i-1 ⊕ L i-1
) for a line bundle L i-1 over X i-1 for all 1 ≤ i ≤ r and r is the dimension of X r . In [START_REF] Choi | Properties of Bott manifolds and cohomological rigidity[END_REF], [START_REF] Choi | Topological classification of generalized Bott towers[END_REF] and [START_REF] Ishida | Filtered cohomological rigidity of Bott towers[END_REF], the authors studied "cohomological rigidity"properties of Bott towers. These also play an important role in algebraic topology and K-theory (see [START_REF] Civan | Homotopy decompositions and k-theory of Bott towers[END_REF], [START_REF] Davis | Convex polytopes, Coxeter orbifolds and torus actions[END_REF] 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 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 • K Xr < 0, where K Xr 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 [START_REF] Anderson | Schubert varieties are log Fano over the integers[END_REF], we say that

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

1 a pair (X, D) of a normal projective variety X and an effective 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 , π i ) : 1 ≤ i ≤ r} is uniquely determined by an upper triangular matrix M r with integer entries, defined via the first Chern class of the line bundle L i-1 on X i-1 , where X i = P(O X i-1 ⊕ L i-1 ) for 1 ≤ i ≤ r (see [GK94, Section 2.3], [START_REF] Civan | Bott towers, crosspolytopes and torus actions[END_REF] and [VT15, Section 7.8]). For more details see Section 2. Let where β ij 's are integers. Define for 1 ≤ i ≤ r, η + i := {r ≥ j > i :

β ij > 0} and η - i := {r ≥ j > i : β ij < 0}. If |η + i | = 1 (respectively, |η + i | = 2), then let η + i = {m} (respectively, η + i = {m 1 , m 2 }). If |η - i | = 1 (respectively, |η - i | = 2), then set η - i = {l} (respectively, η - 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)

|η + i | = 0, |η - i | ≤ 1,
- i | = 1(respectively, |η - i | = 2) then β li = -1 or -2 (respectively, β il 1 = -1 = β il 2 ).
Case 2: If |η - i | = 1 = |η + i | and l < m, then β il = -1, β im = 1 and β mk = 0 for all k > m.

Case 3: Assume that |η + i | = 1. Then β im = 1 and either it satisfies (i) Case 2; or (ii) |η - i | = 0 and β mk = 0 for all k > m; or (iii) there exists a unique r ≥ s > m such that β ms -β is = 1 and β mk -β ik = 0 for all k > s, or β ms -β is = -1 and β is -β ms -β sk = 0 for all k > s.

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 ≤ i ≤ r.

Note that N 1 i =⇒ N 2 i for all 1 ≤ i ≤ r. If X r satisfies condition I, then it also satisfies conditions II. We prove, Theorem (see Theorem 6.3).

(1) X r is Fano if and only if it satisfies I.

(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 Xr denote the tangent bundle of X r .

Corollary (see Corollary 6.4 and Corollary 6.5). If X r satisfies I, then H i (X r , T Xr ) = 0 for all i ≥ 1. In particular, X r is locally rigid.

For 1 ≤ i ≤ 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 ≤ i ≤ r.

Remark 1.2. By using the results of this article, in [START_REF]A note on toric degeneration of a Bott-Samelson-Demazure-Hansen variety[END_REF] 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 [START_REF] Chary | On Fano and weak Fano Bott-Samelson-Demazure-Hansen varieties[END_REF]). We obtain some vanishing theorems for the cohomology of tangent bundle (and line bundles) on BSDH varieties (see also [START_REF]Automorphism group of a Bott-Samelson-Demazure-Hansen variety[END_REF], [CKP] and [START_REF] Chary | Rigidity of a Bott-Samelson-Demazure-Hansen variety for P Sp(2n, C)[END_REF]). We also recover the results in [START_REF] Parameswaran | Toric degeneration of Bott-Samelson-Demazure-Hansen varieties[END_REF].

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.

Preliminaries

In this section we recall toric varieties (see [START_REF] Cox | Toric varieties[END_REF]) and Bott towers (see [START_REF] Civan | Bott towers, crosspolytopes and torus actions[END_REF] and [START_REF] Victor | Toric topology[END_REF]). We work throughout the article over the field 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 [START_REF] Cox | Toric varieties[END_REF] (see also [START_REF] Fulton | Introduction to toric varieties[END_REF] and [START_REF] Oda | Convex bodies and algebraic geometry-An introduction to the theory of toric varieties[END_REF]).

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 = (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 R := M ⊗ R and N R := N ⊗ R. Then we have a natural bilinear pairing -, -

: M R × N R → R. A fan Σ in N R
is a collection of convex polyhedral cones that is closed under intersections and cone faces. Let σ be the dual cone of σ ∈ Σ in M R . For σ ∈ Σ, the semigroup algebra C[σ ∩ M ] is a normal domain and finitely generated C-algebra. Then the scheme Spec(C[σ ∩ M ]) is called the affine toric variety corresponding to σ. For a given fan Σ, we can define a toric variety X Σ by gluing the affine toric varieties Spec(C[σ ∩ M ]) as σ varies in Σ. For all 1 ≤ s ≤ n,

Σ(s) := {σ ∈ Σ : dim(σ) = s}.
For each ρ ∈ Σ(1), we denote u ρ , the generator of ρ ∩ N. For σ ∈ Σ,

σ(1) := Σ(1) ∩ σ.
There is a bijective correspondence between the cones in Σ and the T -orbits in X Σ . For each

σ ∈ Σ, the dimension dim(O(σ)) of the T -orbit O(σ) corresponding to σ is n -dim(σ). Let τ, σ ∈ Σ, then τ is a face of σ if and only if O(σ) ⊂ O(τ ), where O(σ) is the closure of T -orbit O(σ). We denote V (σ) = O(σ)
and it is a toric variety with the corresponding fan being Star(σ), the star of σ which is the set of cones in Σ which have σ as a face. Let D ρ = O(ρ) be the torus-invariant prime divisor in X Σ corresponding to ρ ∈ Σ(1). The group T Div(X Σ ) of T -invariant divisors in X Σ is given by

T Div(X Σ ) = ρ∈Σ(1) ZD ρ .
For each m ∈ M , the character χ m of T is a rational function on X Σ and the corresponding divisor is given by div

(χ m ) = ρ∈Σ(1)
m, u ρ D ρ .

Bott towers.

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

X r = P(O X r-1 ⊕ L r-1 ) X r-1 = P(O X r-2 ⊕ L r-2 )
. . .

X 1 = P(O X 0 ⊕ L 0 ) X 0 = {pt} πr π r-1 π 2 π 1
For each 1 ≤ i ≤ r, X i is a smooth projective toric variety (see [START_REF] Civan | Bott towers, crosspolytopes and torus actions[END_REF]Theorem 22]). Consider the points [1 : 0] and [0 : 1] in P 1 , we call them the south pole and the north pole respectively. The zero section of L i-1 gives a section s 0 i : X i-1 -→ X i , the south pole section; similarly, the north pole section s 1 i : X i-1 -→ X i by letting the first coordinate in P(O X i-1 ⊕ L i-1 ) to vanish.

Let 1 ≤ i ≤ r. Since π i : X i -→ X i-1 is a projective bundle, by a standard result on the cohomology ring of projective bundles we have the following (see [START_REF]Algebraic geometry[END_REF]Page 429] for instance, and also [START_REF] Milne | Etale cohomology (PMS-33)[END_REF]Proposition 10.1]):

Theorem 2.2. The cohomology ring H * (X i , Z) of X i is a free module over H * (X i-1 , Z) on generators 1 and u i , which have degree 0 and 2 respectively, that is

H * (X i , Z) = H * (X i-1 , Z)1 ⊕ H * (X i-1 , Z)u i .
The ring structure is determined by the single relation

u 2 i = c 1 (L i-1
)u i , where c 1 (-) denotes the first Chern class and the restriction of u i to the fiber P 1 ⊂ X i is the first Chern class of the canonical line bundle over P 1 . Hence we have

H * (X i , Z) = H * (X i-1 , Z)[u i ]/J i ,
where J i is the ideal generated by u 2 i -c 1 (L i-1 )u i .

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

0 -→ Z -→ O X i-1 -→ O * X i-1 -→ 0.
Then we get the following exact sequence:

0 -→ H 1 (X i-1 , Z) -→ H 1 (X i-1 , O X i-1 ) -→ H 1 (X i-1 , O * X i-1 ) c 1 (-) -→ H 2 (X i-1 , Z) -→ H 2 (X i-1 , O X i-1 ) -→ • • • Since X i-1 is toric, we have H j (X i-1 , O X i-1 ) = 0 for all j > 0 (see [Oda88, Corollary 2.8]). As H 1 (X i-1 , O * X i-1 ) = P ic(X i-1
), we get c 1 (-) : P ic(X i-1 ) ∼ -→ H 2 (X i-1 , Z). Then we have the following: Theorem 2.3. Each line bundle 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

c 1 (L i-1 ) = - i-1 k=1 β ki u k ∈ H 2 (X i-1 , Z), where β ik 's are integers for 1 ≤ k ≤ i -1.
Then by Theorem 2.2 and 2.3, by iteration, we get the following: Corollary 2.4. We have H * (X r , Z) = Z[u 1 , . . . , u r ]/J, where J is the ideal generated by {u 2 j + i<j β ij u i u j : 1 ≤ j ≤ r} and the integers β ij 's are as in Theorem 2.3.

Write {β ij : 1 ≤ i < j ≤ r}, the collection of r(r -1)/2 integers, as an upper triangular r × r matrix Corollary 2.5. There is a bijective correspondence between {Bott towers of height r} and {r×r upper triangular matrices with integer entries as in (2.1)}.

M r :=       1 β 12 β 13 . . . β
Two Bott towers {(X i , π i ) : 1 ≤ i ≤ r} and {(X i , π i ) : 1 ≤ i ≤ r} are isomorphic if there exists a collection of isomorphisms {φ i : X i → X i : 1 ≤ i ≤ r} such that the following diagram is commutative: Theorem 2.6. The Bott tower {(X i , π i ) :

X r πr / / φr X r-1 π r-1 / / φ r-1 • • • π 2 / / X 1 π 1 / / φ 1 X 0 φ 0 X r π r / / X r-1 π r-1 / / • • • π 2 / / X 1 π 1 / / X
1 ≤ i ≤ r} corresponding to a matrix M r as in (2.1) is isomorphic to {(X Σ i , π Σ i ) : 1 ≤ i ≤ r}
, the collection of smooth projective toric varieties corresponding to the fan Σ i with the 2 i maximal cones generated by the set of vectors {e j : 1 ≤ j ≤ i and ∈ {+, -}}, and where π Σ i :

X Σ i → X Σ i-1 is the toric morphism induced by the projection π Σ i : Z i → Z i-1 for all 1 ≤ i ≤ r.
Note that by Theorem 2.6, Σ i has 2i one-dimensional cones generated by the vectors {e + j , e - j : 1 ≤ j ≤ i}, and by (2.2), we can see that the divisors D ρ + j corresponding to e + j for 1 ≤ j ≤ i form a basis of the Picard group of X i (see Section 3 for more details).

On Picard group of a Bott tower

Now we describe a basis of the Picard group P ic(X r ) of X r . Let ∈ {+, -} and for 1 ≤ i ≤ r, let ρ i be the one-dimensional cone generated by e i . For all 1 ≤ i ≤ r, we define D ρ i to be the toric divisor corresponding to the one-dimensional cone ρ i . We prove, Lemma 3.1. The set {D ρ i : 1 ≤ i ≤ r and ∈ {+, -}} forms a basis of P ic(X r ).

Proof. By Theorem 2.6, using the description of the one-dimensional cones we have the following decomposition of Σ(1):

Σ(1) = {ρ + i : 1 ≤ i ≤ r} ∪ {ρ - i : 1 ≤ i ≤ r}. (3.1)
Again by Theorem 2.6, {D ρ + i : 1 ≤ i ≤ r} forms a basis of the Picard group

P ic(X r ) of X r . Since 0 ∼ div(χ e + i ) = ρ∈Σ(1) u ρ , e + i D ρ ,
by (2.2) we can see that {D ρ - i : 1 ≤ i ≤ r} also forms a basis of P ic(X r ). In general, let σ ∈ Σ be the maximal cone generated by {e i : 1 ≤ i ≤ r}. Take the torus-fixed point x in X r corresponding to the maximal cone σ. Let U be the torus-invariant open affine neighbourhood of x in X r . Then U is an affine space of dimension r; in particular, P ic(U ) = 0. Therefore, we get

X r \ U = ∪ r i=1 D ρ i and P ic(X r ) is generated by {D ρ i : 1 ≤ i ≤ r} (see [Har70, Chapter II, Proposition 3.1, page 66]). Since {D ρ i : 1 ≤ i ≤ r} is linearly independent and the rank of P ic(X r ) is r, this set {D ρ i : 1 ≤ i ≤ r} forms a basis of P ic(X r ).
By Lemma 3.1, the set {D ρ + i : 1 ≤ i ≤ r} forms a basis of P ic(X r ). Now we express for each

1 ≤ i ≤ r, D ρ - i in terms of D ρ + j 's (1 ≤ j ≤ r). Let 1 ≤ i ≤ r, define h i-1 i := -β i(i-1)
and

h j i :=      0 for j > i. 1 for j = i. -i-1 k=j β ik (h j k ) for j < i. Then we prove, Lemma 3.2. Let 1 ≤ i ≤ r. The coefficient of D ρ + j in D ρ - i is h j i .
Proof. Proof is by induction on i and by using

0 ∼ div(χ e + i ) = ρ∈Σ(1) u ρ , e + i D ρ . (3.2)
Recall the equation (2.2),

e - i = -e + i - j>i β ij e + j for all 1 ≤ i ≤ r. If i = 1, by (3.2), we see 0 ∼ div(χ e + 1 ) = D ρ + 1 -D ρ - 1 . Then we have D ρ - 1 ∼ D ρ + 1 . (3.3) If i = 2, by (3.2) and (2.2), we see 0 ∼ div(χ e + 2 ) = D ρ + 2 -D ρ - 2 -β 21 D ρ - 1 . By (3.3), we get D ρ - 2 ∼ D ρ + 2 -β 21 D ρ + 1 = h 2 2 D ρ + 2 + h 1 2 D ρ + 1 . By induction assume that D ρ - k ∼ r j=1 h j k D ρ + j for all k < i.
Again by (3.2) and (2.2), we see

0 ∼ div(χ e + i ) = D ρ + i -D ρ - i - k<i β ik D ρ - k . Then D ρ - i ∼ D ρ + i - k<i β ik D ρ - k . Hence D ρ - i ∼ D ρ + i - k<i β ik ( r j=1 h j k D ρ + j ).
Since h j k = 0 for k < j, we get

D ρ - i ∼ D ρ + i - k<i β ik ( i-1 j=1 h j k D ρ + j ).
Then

D ρ - i ∼ D ρ + i + i-1 j=1 (- i-1 k=j β ik h j k )D ρ + j .
Therefore, we conclude that

D ρ - i ∼ D ρ + i + i-1 j=1 h j i D ρ + j
. This completes the proof of the lemma.

Let ∈ {+, -}. Define Σ(1) :

= {ρ i : 1 ≤ i ≤ r}. Then D = ρ∈Σ(1) a ρ D ρ = ρ∈Σ(1) + a ρ D ρ + ρ∈Σ(1) - a ρ D ρ . For 1 ≤ i ≤ r, let g i := a ρ + i + r j=i a ρ - j h i j . Then we have Corollary 3.3. D = ρ∈Σ(1) a ρ D ρ ∼ r i=1 g i D ρ + i .
Proof. We have

D = ρ∈Σ(1) a ρ D ρ = r i=1 a ρ + i D ρ + i + r i=1 a ρ - i D ρ - i . By Lemma 3.2, we can see that r i=1 a ρ - i D ρ - i ∼ r i=1 a ρ - i ( i j=1 h j i D ρ + j ). Then r i=1 a ρ - i D ρ - i ∼ r i=1 ( r j=i a ρ - j h i j )D ρ + j . Hence we have D ∼ r i=1 (a ρ + i + r j=i a ρ - j h i j )D ρ + i . Thus, D ∼ r i=1 g i D ρ +
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 [START_REF]A note on toric degeneration of a Bott-Samelson-Demazure-Hansen variety[END_REF].

Let 1 ≤ i ≤ r. We prove the following.

Lemma 3.5. The relative tangent bundle T π i of π i : X i → X i-1 is given by

T π i O X i (D ρ + i + D ρ - i ) O X i ( i-1 j=1 β ij D ρ - j + 2D ρ - i ).
Proof. By definition of Bott tower, π i is a P 1 -fibration. Then the relative canonical bundle K π i is given by K

π i = O X i (K X i ) ⊗ π * i (O X i-1 (-K X i-1 )) (see [
K X Σ = - ρ∈Σ(1) D ρ . Then K π i = O X i (- ρ∈Σ(1) D ρ ) ⊗ π * i (O X i-1 ( ρ ∈Σ (1) D ρ )) ,
where Σ is the fan of X i-1 . Since X i-1 smooth, any divisor of the form D = ρ ∈Σ (1) a ρ D ρ with a ρ ∈ Z, in X i-1 is Cartier. Hence the pullback π * i (D) is defined and given by

π * i (D) = π * i ( ρ ∈Σ (1) a ρ D ρ ) = ρ∈Σ(1) -ϕ D (π i (u ρ ))D ρ ,
where ϕ 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

π i : Z i → Z i-1
is the projection onto the first i -1 factors (see page 6), by definition of u ρ and e - j (see (2.2)), for ∈ {+, -} we have

π i (u ρ j ) = u ρ j if 1 ≤ j ≤ i -1. 0 if j = i. Hence -ϕ D (π i (u ρ j )) = a ρ j if 1 ≤ j ≤ i -1. 0 if j = i.
Thus we have,

π * i ( ρ ∈Σ (1) D ρ ) = ρ∈Σ(1)\{ρ + i ,ρ - i } D ρ .
Therefore, we see that

K π i = O X i (-D ρ + i -D ρ - i ). (3.4) By (2.2), we note that 0 ∼ div(χ e + i ) = D ρ + i -D ρ - i - i-1 j=1 β ij D ρ - j . (3.5)
Since Ǩπ i = det T π i , we get Ǩπ i = T π i as π i is a 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 π i can be expressed in terms of

D ρ + i (1 ≤ i ≤ r).
The following is well known and proved here for completeness.

Lemma 3.7. Let X and Y be smooth varieties. Let f : X -→ Y be a fibration with a section σ and denote by σ(Y ) its image in X. Then the restriction of the relative tangent bundle

T f to σ(Y ) is isomorphic to the normal bundle N σ(Y )/X of σ(Y ) in X.
Proof. Consider the normal bundle short exact sequence

0 -→ T σ(Y ) -→ T X | σ(Y ) -→ N σ(Y )/X -→ 0, (3.6)
where T σ(Y ) and T X are the tangent bundles of σ(Y ) and X respectively. Also consider the following short exact sequence

0 -→ T f -→ T X -→ f * T Y -→ 0 .
(3.7)

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

0 -→ T f | σ(Y ) -→ T X | σ(Y ) -→ T σ(Y ) -→ 0 . (3.8)
By using (3.6) and (3.8), we see

T f | σ(Y ) is isomorphic to N σ(Y )/X
. This completes the proof.

We prove,

Lemma 3.8. Let 1 ≤ i ≤ r. The normal bundle N X i /X i-1 of X i-1 in X i is Ľ i-1 , where L i-1 is
V (L ) P(E ) = X i X i-1 π πr s 0 i σπ
Also note that the section s 0 i (X i-1 ) of π i corresponding to the projection E → O X i is same as the zero section σ π (X i-1 ) of π. Now consider the following short exact sequence

0 -→ T π -→ T V (L ) -→ π * T X i-1 -→ 0.
(3.9)

Since the restriction T π|σ π (X i-1 ) of T π to σ π (X r-1 ) is Ľ , by Lemma 3.7 and by above short exact sequence (3.9) we see that N σπ(X i-1 )/V (L )

Ľ . Hence we conclude that N X i-1 /X i Ľ (here we are identifying X i-1 with the section corresponding to the projection

E = O X i-1 ⊕L → O X i-1
). This completes the proof of the lemma.

Let 1 ≤ i ≤ r. We prove, Lemma 3.9.

(1) The toric sections of π i are given by D ρ i , ∈ {+, -}.

(2) The normal bundle

N X i-1 /X i of X i-1 in X i is given by N X i-1 /X i = Ľ i-1 = O X i (D ρ + i )
, where the line bundle L i-1 is as in the definition of the Bott tower X i .

Proof. Proof of (1): Recall that π i is a P 1 -fibration induced by the projection π i : Z i → Z i-1 . For each cone σ ∈ Σ F of dimension 1 (which is a maximal cone in Σ F , where Σ F denote the fan of the fiber P 1 ), the subvariety V (σ) is an invariant section of π i , which is an invariant divisor in X i . Hence we get two invariant divisors

V (ρ + i ) = D ρ + i and V (ρ - i ) = D ρ - i .
Proof of (2): By Lemma 3.8, we have N X i-1 /X i = Ľ i-1 and the section X i-1 is given by the projection

E = O X i-1 ⊕ L i-1 → 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 Σ, which are basic tools for the classification of Fano toric varieties due to Batyrev (see [START_REF] Batyrev | On the classification of smooth projective toric varieties[END_REF]).

Definition 4.1. We say P ⊂ Σ(1) is a primitive collection if P is not contained in σ(1) for some σ ∈ Σ but any proper subset is. Note that if Σ is simplicial, primitive collection means that P does not generate a cone in Σ but every proper subset does. Definition 4.2. Let P = {ρ 1 , . . . , ρ k } be a primitive collection in a complete simplicial fan Σ. Recall u ρ is the primitive vector of the ray ρ ∈ Σ. Then k i=1 u ρ i is in the relative interior of a cone γ P in Σ with a unique expression

k i=1 u ρ i = ρ∈γ P (1) c ρ u ρ , c ρ ∈ Q >0 . Hence we have k i=1 u ρ i -( ρ∈γ P (1) c ρ u ρ ) = 0. (4.1)
Then we call (4.1) the primitive relation of X Σ corresponding to P.

Recall that T Div(X Σ ) denote the group of torus-invariant divisors in X Σ (see Page 4). Since the fan Σ of X r is full dimensional, we have the following short exact sequence

0 -→ M ϕ 1 -→ T Div(X r ) = ρ∈Σ(1) ZD ρ ϕ 2 -→ P ic(X r ) → 0, (4.2)
where the maps are given by ϕ 1 : m → div(χ m ) and ϕ 2 :

D → O Xr (D) (see [CLS11, Theorem 4.2.1]).
Now we recall some standard notations: Let X be a smooth projective variety, we define

N 1 (X) Z := { finite a i C i : a i ∈ Z, C i irreducible curve in X}/ ≡
where ≡ is the numerical equivalence, i.e. Z ≡ Z if and only if

D • Z = D • Z for all divisors D in X. We denote by [C] the class of C in N 1 (X) Z . Let N 1 (X) := N 1 (X) Z ⊗ 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) Z is dual to P ic(X) via the natural pairing (see [CLS11, Proposition 6.3.15]). In our case X = X r , there are dual exact sequences:

0 -→ M ϕ 1 -→ Z Σ(1) ϕ 2 -→ P ic(X r ) -→ 0 and 0 -→ N 1 (X r ) Z ϕ * 2 -→ Z Σ(1) ϕ * 1 -→ N -→ 0, (4.3) where ϕ * 2 ([C]) = (D ρ • C) ρ∈Σ(1) ,
C is an irreducible complete curve in X r and ϕ * 1 (e ρ ) = u ρ , e ρ is a standard basis vector of R Σ(1) (see [CLS11, Proposition 6.4.1]). Let P be a primitive collection in Σ. Note that since X r is smooth projective, P ∩ γ P (1) = ∅ and c ρ ∈ Z >0 for all ρ ∈ γ P (1) (4.4)

(see [CLS11, Proposition 7.3.6]). As an element in Z (1) , we write r(P ) = (r ρ ) ρ∈Σ(1) , where

r ρ =      1 if ρ ∈ P -c ρ if ρ ∈ γ P (1) 0 otherwise (4.5)
Then by (4.1) we see that

ρ∈ (1)
r ρ u ρ = 0.
Hence by the exact sequence (4.3) and by (4.4), we observe that r(P ) gives an element in N 1 (X r ) Z (see [CLS11, Page 305]). We prove, Lemma 4.3.

Let P i := {ρ + i , ρ - i }, 1 ≤ i ≤ r. Then {P i : 1 ≤ i ≤ r} is the set of all primitive collections of the fan Σ of X r .
Proof. By Theorem 2.6, the cones in the fan Σ of X r are generated by subsets of {e + 1 , . . . , e + r , e - 1 , . . . , e - r } and containing no subset of the form {e + i , e - i }. Then by Definition 4.1, it is clear that P i = {ρ + i , ρ - i } is a primitive collection for all i. Also note that again by description of the cones in Σ, any primitive collection must contain a P i for some 1 ≤ i ≤ r.

Fix 1 ≤ i ≤ r. Let Q be a collection of one-dimensional cones such that it properly contains P i , i.e. there exists 1 ≤ j ≤ r and j = i such that ρ j ∈ Q ⊃ P i , ∈ {+, -}. Assume that Q is a primitive collection. Then by Definition 4.1, {ρ + i , ρ - i } ⊂ Q generates a cone in Σ. This is a contradiction to the description of the cones in Σ. Therefore, we conclude that {P i : 1 ≤ i ≤ r} is the set of all primitive collections. Now we define the Contractible classes from [START_REF] Casagrande | Contractible classes in toric varieties[END_REF]: Let X be a smooth projective toric variety. We define N E(X) Z in N 1 (X) by

N E(X) Z := { f inite a i C i : a i ∈ Z ≥0 and C i irreducible curve in X }.
Let γ ∈ N E(X) Z be primitive (i.e. the generator of Z ≥0 γ) and such that there exists some irreducible curve in X having numerical class in Q ≥0 γ. Then Definition 4.4. (see [Cas03, Definition 2.3]) The above class γ is called contractible if there exists a toric variety X γ and an equivariant morphism φ γ : X → X γ , surjective with connected fibers, such that for every irreducible curve C in X, Proposition 4.6. Let P = {ρ 1 , . . . , ρ k } be a primitive collection in Σ, with the primitive relation r(P ):

φ γ (C) = {pt} if and only if [C] ∈ Q ≥0 γ.
k i=1 u ρ i - ρ∈γ P (1) c ρ u ρ = 0.
Then r(P ) is contractible if and only if for every primitive collection Q of Σ such that P ∩Q = ∅ and P = Q, the set (Q \ P ) ∪ γ P (1) contains a primitive collection.

Mori cone.

We use the notation as above. Let X be a smooth projective variety. We define N E(X) the real convex cone in N 1 (X) generated by classes of irreducible curves. The Mori cone N E(X) is the closure of N E(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 N E(X) Z is generated by the finitely many torus-invariant irreducible curves in X and hence N E(X) Z is a finitely generated monoid. Hence the cone N E(X) = N E(X) is a rational polyhedral cone and we have

N E(X) = τ ∈Σ(r-1) R ≥0 [V (τ )],
where r = dim(X) and [V (τ )] ∈ N 1 (X) Z is the class of the toric curve V (τ ). This is called the Toric Cone Theorem (see [START_REF] Cox | Toric varieties[END_REF]Theorem 6.3.20]). Let τ ∈ Σ(r -1) be a wall, that is τ = σ ∩ σ for some σ, σ ∈ Σ(r). Let σ (respectively, σ ) is generated by {u ρ 1 , u ρ 2 , . . . , u ρr } (respectively, by {u ρ 2 , . . . , u ρ r+1 }) and let τ be generated by {u ρ 2 , . . . , u ρr }. Then we get a linear relation,

u ρ 1 + r i=2 b i u ρ i + u ρ r+1 = 0 (4.6)
The relation (4.6) called wall relation and we have

D ρ • V (τ ) =      b i if ρ = ρ i and i ∈ {2, 3, . . . , r} 1 if ρ = ρ i and i ∈ {1, r + 1} 0 otherwise
(see [CLS11, Proposition 6.4.4 and eq. (6.4.6) page 303]). Now we describe the Mori cone N E(X r ) of X r in terms of the primitive relations of X r .

Theorem 4.7. N E(X r ) Z = r i=1 Z ≥0 r(P i ).

Proof. We have N E(X r ) = P ∈P R ≥0 r(P ) , where P is the set of all primitive collections in X r (see [CLS11, Theorem 6.4.11]). By Lemma 4.3, {P i : 1 ≤ i ≤ r} is the set of all primitive collections of X r . Therefore, we get

N E(X r ) = r i=1 R ≥0 r(P i ) .
By [Cas03, Theorem 4.1], we have

N E(X r ) Z = γ∈C Z ≥0 γ,
where 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 ≤ i ≤ r. Since any contractible class is a primitive relation, we get

C = {r(P i ) : 1 ≤ i ≤ r}.
Hence we conclude that

N E(X r ) Z = r i=1 Z ≥0 r(P i ).
This completes the proof of the theorem.

We have

Corollary 4.8. The set {r(P i ) : 1 ≤ i ≤ r} forms a basis of N 1 (X r ) Z .

Proof. By Theorem 4.7, {r(P i ) : 1 ≤ i ≤ r} generates the monoid N E(X r ) Z and the cone N E(X r ) is of dimension r. So r(P i ) for 1 ≤ i ≤ r are linearly independent. Also the group N 1 (X r ) Z is generated by N E(X r ) Z , hence by r(P i ) for 1 ≤ i ≤ r. Hence these form a basis of N 1 (X r ) Z .

Next we describe the primitive relation r(P i ) explicitly by finding the cone γ P i in (4.1) for 1 ≤ i ≤ 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 (1) Let r ≥ j > j 1 = i ≥ 1 and define a 1,j := β j 1 j .

M r =       1 β 12 β 13 . . .
(2) Let r ≥ j 2 > j 1 be the least integer such that a 1,j > 0, then define for j > j 2 a 2,j := β ij 2 β j 2 j -β ij .

(3) Let k > 2 and let r ≥ j k > j k-1 be the least integer such that a k-1,j < 0, then inductively, define for j > j k a k,j := -a k-1,j k β j k j + a k-1,j .

(4) For j ≤ i, b j := 0, and for j > i define b j := a l,j if j l+1 ≥ j > j l , l ≥ 1. (4.7)

Note that we have Then j 3 = 6 and a 3,7 = -a 2,6 β 67 + a 2,7 = -(-1)(-1) + (2) = 1.

b j =      0 for j ≤ i < 0 for j ∈ {j 3 , . . . , j m } ≥ 0 otherwise . (5) Let I i := {j 1 , . . . , j m }. Example 4.9. Let M 7 =          1 -1 -1 -1 2 -1 2 0 1 0 2 -1 2 -1 0 0 1 0 -1 -1 -1 0 0 0 1 -1 2 -1 0 0 0 0 1 -1 2 0 0 0 0 0 1 -1 0 0 0 0 0 0 1          7×7 Let i = 1,
Therefore, I 1 = {1, 5, 6} . Let 1 ≤ i ≤ r. Let A i := {e j j : 1 ≤ j ≤ r, b j = 0 and j = + for j / ∈ I i -for j ∈ I i }.
Remark 4.10. Note that as b j = 0 for j ≤ i, we can take i < j ≤ r in the definition of A i . Proof. By (2.2), for all 1 ≤ i ≤ r, we have

Now

e - i + e + i = - j>i β ij e + j .
(4.9)

If for all j > i, β ij ≤ 0, then the cone γ P i is generated by {e + j : j > i, β ij < 0}. If not, choose the least integer j 2 > i such that β ij 2 > 0. Now write

e - i + e + i = -( j 2 >j>i β ij e + j ) + β ij 2 (-e + j 2 ) -( j>j 2 β ij e + j ).
Again by using (4.9), we have

e - i + e + i = -( j 2 >j>i β ij e + j ) + β ij 2 (e - j 2 + j>j 2 β j 2 j e + j ) -( j>j 2 β ij e + j )
.

Then e - i + e + i = -(

j 2 >j>i β ij e + j ) + β ij 2 e - j 2 + j>j 2 (β ij 2 β j 2 j -β ij )e + j )
.

By definition a 2,j = β ij 2 β j 2 j -β ij , then we have

e - i + e + i = -( j 2 >j>i β ij e + j ) + β ij 2 e - j 2 + ( j>j 2 a 2,j e + j ).
If a 2,j ≥ 0 for all j > j 2 , then γ P i is generated by {e j j : j > i, j = +∀j = j 2 , and j = -for j = j 2 }.

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

e - i + e + i = -( j 2 >j>i a 1,j e + j ) + β ij 2 e - j 2 + ( j 3 >j>j 2 a 2,j e + j ) -a 2,j 3 (e - j 3 + j>j 3 β j 3 j e + j ) + ( j>j 3 a 2,j e + j ).
Then,

e - i + e + i = -( j 2 >j>i a 1,j e + j ) + β ij 2 e - j 2 + ( j 3 >j>j 2 a 2,j e + j ) -a 2,j 3 e - j 3 + j>j 3
(-a 2,j 3 β j 3 j + a 2,j )e + j ).

By definition a 3,j = -a 2,j 3 β jj 3 + a 2,j , then we have

e - i + e + i = -( j 2 >j>i a 1,j e + j ) + 2e - j 2 + ( j 3 >j>j 2 a 2,j e + j ) -a 2,j 3 e - j 3 + ( j>j 3 a 3,j e + j ).
By repeating this process, we get the cone γ P i as we required.

Let 1 ≤ i ≤ r. Recall I i = {i = j 1 , . . . , j m } as in page 13. Define for 1 ≤ j ≤ r ,

c j := -b j if j ∈ I i \ {j 1 , j 2 } b j otherwise
Set γ P i (1) := {γ 1 , . . . , γ l }. Then we have Corollary 4.13. For 1 ≤ i ≤ r, the primitive relation r(P i )(= (r ρ ) ρ∈Σ(1) ) of X r given by

r ρ =      1 for ρ = ρ + i or ρ - i -c j for ρ = γ j ∈ γ P i (1) 0 otherwise
Example 4.14. We use Example 4.12. The following can be seen easily from (4.8).

(1)

γ P 1 (1) = {ρ + 2 , ρ + 3 , ρ + 4 , ρ - 5 , ρ - 6 , ρ + 7 } .
(2) The primitive relation r(P 1 ) = (r ρ ) ρ∈Σ( 1) is given by

r ρ =          1 for ρ = ρ + 1 or ρ - 1 -1 for ρ = ρ + k , k ∈ {2, 3, 4, 7} and ρ = ρ - 6 -2 for ρ = ρ - 5 0 otherwise
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 ≤ i ≤ r. Let C i := {e j j : 1 ≤ j ≤ r and

j = + if j / ∈ I i \ {j 1 } -if j ∈ I i }.
Let C i := {e j j : 1 ≤ j ≤ r and

j = + if j / ∈ I i -if j ∈ I i }.
Example 4.15. We use Example 4.12, for i = 1, we have I 1 = {1, 5, 6}. Then

C 1 = {e + 1 , e +
2 , e + 3 , e + 4 , e - 5 , e - 6 , e + 7 } and C 1 = {e - 1 , e + 2 , e + 3 , e + 4 , e - 5 , e - 6 , e + 7 }.

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

Proposition 4.16. Fix 1 ≤ i ≤ r. The class of curve r(P i ) is given by

r(P i ) = [V (τ i )] ,
where τ i = σ ∩ σ and σ (respectively, σ ) is the cone generated by C i (respectively, by C i ).

Proof. From Corollary 4.13, we have the following.

e + i + e - i -j>i c j e j j = 0, (4.10)

where j is as in Proposition 4.11. First we show that the set

Q := {ρ ∈ Σ(1) : D ρ • V (τ i ) > 0}
is not contained in σ(1) for any σ ∈ Σ (we adapt the arguments of [CLS11, Proof of Theorem 6.4.11, page 306], here we are not assuming the curve V (τ i ) is extremal). Indeed, suppose Q ⊆ σ(1) for some σ ∈ Σ. 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

D = ρ∈Σ(1)
a ρ D ρ , a ρ = 0 for all ρ ∈ σ(1) and a ρ ≥ 0 for all ρ / ∈ σ(1)

(see [START_REF] Cox | Toric varieties[END_REF](6.4.10), page 306]). Then we can see

D • V (τ i ) = ρ / ∈σ(1) a ρ D ρ • V (τ i ). As Q ⊆ σ(1), by definition of Q, D ρ • V (τ i ) ≤ 0 for ρ / ∈ σ(1). Since a ρ ≥ 0 for ρ / ∈ σ(1), we get D • V (τ i ) ≤ 0,
which is a contradiction as D is ample. Therefore, Q is not contained in σ(1) for any σ ∈ Σ. Hence to prove the proposition it is enough to prove

P i = Q(:= {ρ ∈ Σ(1) : D ρ • V (τ i ) > 0})
(see again [CLS11, Proof of Theorem 6.4.11, page 306]). From (4.10) and by using wall relation, we can see that

D ρ • V (τ i ) =      1 if ρ = ρ + i or ρ - i . -c j if ρ = ρ j j and j ∈ I i \ {j 1 }. 0 otherwise.
Since c j 's are all positive integers (see (4.4)), by Lemma 4.3 we conclude that

P i = {ρ ∈ Σ(1) : D ρ • V (τ i ) > 0} and hence r(P i ) = [V (τ i )]
. This completes the proof of the proposition.

Example 4.17. In Example 4.12, the curve r(P 1 ) = [V (τ 1 )] with τ 1 = σ ∩ σ where σ is the cone generated by

C 1 = {e + 1 , e +
2 , e + 3 , e + 4 , e - 5 , e - 6 , e + 7 } and σ is the cone generated by

C 1 = {e -
1 , e + 2 , e + 3 , e + 4 , e - 5 , e - 6 , e + 7 }.

Corollary 4.18. N E(X r ) Z = r i=1 Z ≥0 [V (τ i )]
, where τ i is as in Proposition 4.16.

Proof. This follows from Theorem 4.7 and Proposition 4.16

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 N ef (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 N ef (X) (see [Kle66, Theorem 1, §2, Chapter IV]). Recall that the nef cone N ef (X) and the Mori cone N E(X) are closed convex cones and are dual to each other (see [Kle66, §2, Chapter IV] ) .

In our case, we have P ic(X r ) 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 = a ρ D ρ be a toric divisor in X r and for 1 ≤ i ≤ r, define

d i := (a ρ + i + a ρ - i - γ j ∈γ P i (1) c j a γ j ).
Then we prove, Lemma

(1) The divisor D is ample if and only if d i > 0 for all 1 ≤ i ≤ r.

(2) The divisor D is numerically effective (nef) if and only if d i ≥ 0 for all 1 ≤ i ≤ r.

Proof. Proof of (2): Recall that the primitive relation r(P i ) is given by

r(P i ) = (r ρ ) ρ∈Σ (1)
(see page 11). First observe that we have the following

D • r(P i ) = ρ∈Σ(1) a ρ (D ρ • r(P i )) = ρ∈Σ(1)
a ρ r ρ (see [CLS11, Proposition 6.4.1, page 299]). Then by (4.5), we get

D • r(P i ) = ρ∈P i a ρ - ρ∈γ P i (1)
r ρ a ρ .

By Lemma 4.3, we have

P i = {ρ + i , ρ - i }.
Then by Corollary 4.13, we get

D • r(P i ) = (a ρ + i + a ρ - i - γ j ∈γ P i (1)
c j a γ j ) =: d i .

(5.1)

Since the nef cone N ef (X r ) and the Mori cone N E(X r ) are dual to each other, the divisor D is nef if and only if D • C ≥ 0 for all torus-invariant irreducible curves C in X r . By Theorem 4.7, we have

N E(X r ) = r i=1 R ≥0 r(P i ).
Hence D is nef if and only if D • r(P i ) ≥ 0 for all 1 ≤ i ≤ r. Therefore, by (5.1), we conclude that the divisor D is nef if and only if d i ≥ 0 for all 1 ≤ i ≤ r. This completes the proof of (2).

Proof of (1): Recall that the divisor D is ample if and only if its class in P ic(X r ) R lies in the interior of the nef cone N ef (X r ). Hence by using similar arguments as in the proof of (2) and the toric Kleiman criterion for ampleness [START_REF] Cox | Toric varieties[END_REF]Theorem 6.3.13], we can see that D is ample if and only if d i > 0 for all 1 ≤ i ≤ r.

Next we describe the generators of the nef cone N ef (X r ) of X r .

Example 5.2. Let M 2 = 1 -1 0 1 2×2 . Then X 2 = P(O P 1 ⊕ O P 1 (1)), the Hirzebruch surface H 1 and the rays ρ + 1 , ρ - 1 , ρ + 2 and ρ - 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.

ρ + 1 ρ + 2 ρ - 1 ρ - 2 Figure. Fan of Hirzebruck surface H 1 .
The primitive relations r(P 1 ) and r(P 2 ) are given by r(P 1 ) : e + 1 + e - 1 = e 2 and r(P 2 ) : e + 2 + e - 2 = 0. By wall relation, we observe that

(1) D ρ + 1 • r(P 1 ) = 1 and D ρ + 1 • r(P 2 ) = 0.
(2) D ρ - 2 • r(P 1 ) = 0 and D ρ - 2 • r(P 2 ) = 1. Then the dual basis of {r(P 1 ) , r(P

2 )} is {D ρ + 1 , D ρ - 2 }.
D 7 = D 1 + D 2 + D 3 + D 6 + D ρ + 7 .
We prove, Proposition 5.6. The set {D i : 1 ≤ i ≤ r} is dual basis of {r(P i ) : 1 ≤ i ≤ r}.

Proof. Fix 1 ≤ i ≤ r. By Proposition 4.16, the class of curve corresponding to the primitive relation r(P i ) is given by r(P i ) = [V (τ i )] (where τ i is described as in Proposition 4.16). From Corollary 4.13, the primitive relation

r(P i )(= [V (τ i )]) is e + i + e - i - j>i c j e j j = 0, (5.2)
where j is as in Proposition 4.16. Note that this is the wall relation for the torus-invariant curve V (τ i ). We prove

D m • r(P i ) = D m • V (τ i ) = 1 if i = m. 0 if i = m.
(5.3) By (5.2) and by wall relation, we have

D ρ + m • V (τ i ) =          1 for m = i 0 for m < i -c γ P i ρ + m for m > i and i ∈ J m 0 for m > i and i / ∈ J m (5.4)
Hence by definition of D m , it is clear that

D m • V (τ i ) = 1 for m = i 0 for m < i (5.5)
Now we claim D m • V (τ i ) = 0 for all m > i. Assume that m > i and write m = i + j, where 1 ≤ j ≤ 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 = ∅, then D i+1 = D ρ + i+1 . By (5.4), we see that D i+1 • V (τ i ) = 0. Case 2: Assume that J i+1 = ∅.

Subcase 1: If i / ∈ J i+1 , then by (5.4) and (5.5), we can see that

D i+1 • V (τ i ) = 0.
Subcase 2: If i ∈ J i+1 , then by (5.5), we have

D i+1 • V (τ i ) = c γ P i ρ + i+1 + (D ρ + i+1 • V (τ i )). By (5.4), D ρ + i+1 • V (τ i ) = -c γ P i ρ + i+1
and hence D i+1 • V (τ i ) = 0. This proves the claim for j = 1. Now assume that j > 1.

Case 1: If J m = ∅, then by (5.4) and (5.5), we see that

D m • V (τ i ) = 0.
Case 2: Assume that J m = ∅.

Subcase 1: If i / ∈ J m , then by (5.4) and (5.5), we can see that

D m • V (τ i ) = (( k∈Jm,k>i c γ P k ρ + m D k ) • V (τ i )) + (D ρ + m • V (τ i )).
By induction on j, D k • V (τ i ) = 0 for all i < k < m. By (5.4), as m > i and m / ∈ J m , we have

D ρ + m • V (τ i ) = 0 . Hence we conclude that D m • V (τ i ) = 0.
This completes the proof of the proposition.

We have, Theorem 5.7.

(1) The nef cone N ef (X r ) of X r is generated by {D i : 1 ≤ i ≤ r}.

(2) The divisor D = i a i D i is ample if and only if a i > 0 for all 1 ≤ i ≤ r.

Proof. Since the nef cone N ef (X r ) is dual of the Mori cone N E(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 N ef (X r ).

Fanoness and weak Fanoness of Bott towers

In this section we describe the matrices M r such that the corresponding Bott tower X r is Fano or weak Fano. Observe that κ(D) ∈ {-∞, 0, 1, . . . , dim(X)}. We say D is big if κ(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 \ U . Then D is big.

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

There exists f 1 , . . . , f n ∈ O X (U ) algebraically independent over C, where n = dim(X). View f 1 , . . . , f n as rational functions on X, then f 1 , . . . , f n ∈ H 0 (X, O X (E)) for some effective divisor E with support X \ U (since div(f i ) is an effective divisor with support in X \ U for 1 ≤ i ≤ r). Thus, the monomials in f 1 , . . . , f n of any degree m are linearly independent elements of H 0 (X, O X (mE)). So dim(H 0 (X, O X (mE))) grows like m n as m → ∞. 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 \ 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) X r is Fano if and only if it satisfies I.

(2) X r is weak Fano if and only if it satisfies II. By using Lemma 5.1, we prove that -K Xr is nef if and only if X r satisfies II.

Let D = -K Xr . By (6.1) and by definition of d i for D (see Lemma 5.1), we have

d i = 2 - γ j ∈γ P i (1) c j .
Then by Lemma 5.1(2), -K Xr is nef if and only if γ j ∈γ P i (1) c j ≤ 2 for all 1 ≤ i ≤ r.

First assume that -K Xr is nef. Fix 1 ≤ i ≤ r. By above discussion, we have

γ j ∈γ P i (1) c j ≤ 2. (6.2)
Since c j 's are positive integers (see (4.4)), we get the following situation: Hence X r satisfies the condition N 2 i . Case 3: If |γ P i (1)| = 2, then there exists r ≥ s 1 > s 2 > i with γ s 1 , γ i | = 1 and β is 1 = 1. Then β s 1 s 2 -β is 2 = -1 and β is 2 -β s 1 s 2 -β s 2 k = 0 for all k > s 2 . Hence X r satisfies the condition N 2 i . 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).

|γ P i (1)| = 0 or |γ P i (1)| = 1, or |γ P i (1)| = 2.
Proof of (1): This follows by using similar arguments as in the proof of (2) and Lemma 5.1(1). 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 Xr denotes the tangent bundle of X r . Then we have Corollary 6.4. If X w satisfies I, then H i (X w, T X w ) = 0 for all i ≥ 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 Xr ) = 0 for all i ≥ 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 6.5. The Bott tower X r is locally rigid if it satisfies I.

Log Fanoness of Bott towers

Recall that a pair (X, D) of a normal projective variety X and an effective Q-divisor D is Kawamata log terminal (klt) if K X + D is Q-Cartier, and for all proper birational maps f : Y -→ X, the pull back f * (K X + D) = K Y + D satisfies f * K Y = K X and D ≤ 0, where i a i D i = i a i D i , x is the greatest integer ≤ 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 [START_REF] Cascini | On the anti-canonical ring and varieties of Fano type[END_REF]). Let X be a Q-Gorenstein projective normal variety over C. If X admits a divisor D with the pair (X, D) being log Fano then -K X is big (In [START_REF] Cascini | On the anti-canonical ring and varieties of Fano type[END_REF] 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 D = 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 [START_REF] Cox | Toric varieties[END_REF]Definition 11.4.23 and Proposition 11.4.24, page 558]. We use notation as in Lemma 5.1. Let D = ρ∈Σ(1) a ρ D ρ be a toric divisor in X r , with a ρ s in Q ≥0 and D = 0. For 1 ≤ i ≤ r, define

k i := d i -2 + γ j ∈γ P i (1) c j .
Then we prove, Theorem 7.1. The pair (X r , D) is log Fano if and only if k i < 0 for all 1 ≤ i ≤ r.

  and if |η - i | = 1 then β il = -1; or (ii) |η - i | = 0, |η + i | ≤ 1, and if |η + i | = 1 then β im = 1 and β mk = 0 for all k > m. • N 2 i is the condition that Case 1: Assume that |η + i | = 0. Then |η - i | ≤ 2, and if |η

  as in the definition of Bott tower and Ľ i-1 is denotes the dual of L i-1 . Proof. Fix 1 ≤ i ≤ r and let L := L i-1 . Recall that P(E ) is by definition P roj(S(E )), S(E ) is symmetric algebra of E = O X i-1 ⊕ L (see [Har77, Page 162]). Let V (L ) = Spec(S(L )), the geometric vector bundle associated to the locally free sheaf (line bundle) L (see [Har77, Exercise 5.18, Page 128]). Then, V (L ) is an open subvariety in P(E ) and we have the following commutative diagram

Remark 4. 5 .

 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].

  2). Fix 1 ≤ i ≤ r. Define:

  then j 1 = 1 and (1) a 1,2 = β 12 = -1 ; (2) a 1,3 = β 13 = -1 ; (3) a 1,4 = β 14 = -1 ; (4) a 1,5 = β 15 = 2 ; (5) a 1,6 = β 16 = -1 ; (6) a 1,7 = β 17 = 2. Then j 2 = 5 and (1) a 2,6 = β 15 β 56 -β 16 = -1 ; (2) a 2,7 = β 15 β 57 -β 17 = 2 .

( 5 )D 5 =

 55 If m = 5, then J 5 = {3, 4} and c D 3 + D 4 + D ρ + 5 . (6) If m = 6, then J 6 = {2, 5} and c Hence D 6 = D 2 + D 5 + D ρ + 6 . (7) If m = 7, then J 7 = {1, 2, 3, 6} and c

  First recall the Iitaka dimension of a Cartier divisor D in a normal projective variety X. Let N (D) := {m ≥ 0 : H 0 (X, L (mD)) = 0}, where L (mD) is the line bundle associated to mD. For m ∈ N (D), we have a rational map φ m : X P(H 0 (X, L (mD)) * ). If N (D) is empty we define the Iitaka dimension κ(D) of D as -∞. Otherwise we define κ(D) := max m∈N (D){dim(φ m (X))}.

Proof.

  Proof of (2): We haveK Xr = -CLS11, Theorem 8.2.3] or[START_REF] Fulton | Introduction to toric varieties[END_REF] Page 74]). The anti-canonical line bundle of any projective toric variety is big, since we havesupp(-K Xr ) = X r \ (C * ) r , (C * ) r isan affine open subset of X r , by Corollary 6.2, -K Xr is big.

  + i | = 0 and c j = -β ij = 1. Then β ij = -1 and hence X r satisfies the condition N 1 i . If the primitive relation is (6.4), then by (2.2) |η - i | = 0 and |η + i | = 1. Hence c j = β ij = 1 and β jk = 0 for all k > j. Subcase (ii): Assume that c j = 2. If the primitive relation r(P i ) is (6.3), then |η + i | = 0 and |η - i | = 1. So by (2.2), we have c j = -β ij . If the primitive relation r(P i ) is (6.4), then |η + i | = 1, |η - i | = 0 and β jk = 0 for all k > j. Again by (2.2), we have c j = β ij Thus, either β ij = -2 or β ij = 2.

Proof.

  From the above discussion by the condition on D, the pair (X r , D) is log Fano if and only if -(K Xr + D) is ample. Note that as -K Xr = ρ∈Σ(1) D ρ , we get-(K Xr + D) = ρ∈Σ(1) (1 -a ρ )D ρ . By Lemma 5.1, -(K Xr + D) is ample if and only if ((1 -a ρ + i ) + (1 -a ρ - i )γ j ∈γ P i (1)c j (1 -a γ j )) > 0 for all 1 ≤ i ≤ r. (7.1)

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

  we have, Proposition 4.11. Let 1 ≤ i ≤ r. The cone γ P i in the primitive relation of X r corresponding to P i is generated by A i . Now we describe the cone γ P 1 . Observe that in (i) coefficient of e +

	5 is negative. By (v), we
	can see
	e -1 + e + 1 = e + 2 + e + 3 + e + 4 + 2(e -5 -e + 6 + 2e + 7 ) + e + 6 -2e + 7 .
	Then e -1 + e + 1 = e + 2 + e + 3 + e + 4 + 2e -5 -e + 6 + 2e + 7 . By (vi),
	e -1 + e + 1 = e + 2 + e + 3 + e + 4 + 2e -5 + e -6 + e + 7 .	(4.8)
	In this case, I 1 = {1, 5, 6} (see Example 4.9) and the cone γ P 1 is generated by
	{e + 2 , e + 3 , e + 4 , e -5 , e -6 , e + 7 }.
	Now we prove Proposition 4.11:

  Hence the generators of the nef cone N ef (H 1 ) are D ρ + 1 and D ρ - 2 . Note that by Lemma 3.1, P ic(H 1 ) is generated by {D ρ + 1 , D ρ - 2 }.

  Case 1: If |γ P i (1)| = 0, then by definition of γ P i (see Definition 4.2), we have Hence we see X r satisfies the condition N 1 i . Case 2: If |γ P i (1)| = 1, then there exists a unique r ≥ j > i, such that γ j ∈ γ P

	r(P i ) : e + i + e -i = 0.
	Then |η + i | = 0 = |η -i |. i (1) and the
	primitive relation is either
	r(P i ) : e + i + e -i = c j e + j	(6.3)
	or
	r(P i ) : e + i + e -i = c j e -j	(6.4)
	By (6.2), we get c j = 1 or 2.

Subcase (i): Assume that c j = 1. If the primitive relation is (6.3), then we can see that |η

  s 2 ∈ γ P i (1) such that the primitive relation r(P i ) is r(P i ) :e + i + e - i = c s 1 e ± s 1 + c s 2 e ± Ifthe primitive relation is r(P i ) : : e + i + e - i = c s 1 e + s 1 + c s 2 e + s 2 , by (2.2) we see |η + i | = 0 and |η - i | = 2 . By (6.2) and (4.4) (c i 's are positive integers), we getc s 1 = 1 , c s 2 = 1 and β is 1 = β is 2 = -1.Hence X r satisfies the conditionN 2 i . Subcase (ii): If the primitive relation is r(P i ) : e + i + e - i = c s 1 e + s 1 + c s 2 e - s 2 , by (2.2) we see |η + i | = 1 = |η - i |. Then β is 1 = -1, β is 2 = 1 and β s 2 k = 0 for all k > s 2 . Subcase (iii): If the primitive relation is r(P i ) : e + i + e - i = c s 1 e - s 1 + c s 2 e + s 2 , by (2.2) we see |η + i | = 1 and β is 1 = 1. Then β s 1 s 2 -β is 2 = 1 and β s 1 k -β ik = 0 for all k > s 2 . Subcase(iv): If the primitive relation is r(P i ) : e + i + e - i = c s 1 e - s 1 + c s 2 e - s 2 , by (2.2) we see |η +

	s 2	(6.5)
	Subcase (i):

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

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 Now we prove the similar results for X r . For 1 ≤ m ≤ r, define

Note that the set J m is the collection of indices i < m for which u ρ + m appear in the γ P i part of the expression (4.1) for the primitive relation r(P i ).

We set D 1 := D ρ + 1 , and for m > 1 define inductively

where -c

(1) Recall by (4.8), we have e - 1 + e + 1 = e + 2 + e + 3 + e + 4 + 2e - 5 + e - 6 + e + 7 . Then,

(2) γ P 2 (1) = {ρ - 4 , ρ - 5 , ρ + 6 , ρ + 7 } (since e + 2 + e - 2 = 2e - 4 + e - 5 + e + 6 + e + 7 ) .

(3) γ P 3 (1) = {ρ + 5 , ρ + 7 } (since e + 3 + e - 3 = e + 5 + e + 7 ) .

(4) γ P 4 (1) = {ρ + 5 , ρ - 6 , ρ - 7 } (since e + 4 + e - 4 = e + 5 + 2e - 6 + e - 7 ).

(5) γ P 5 (1) = {ρ + 6 , ρ - 7 } (since e + 5 + e - 5 = e + 6 + 2e - 7 ).

(6) γ P 6 (1) = {ρ + 7 } (since e + 6 + e - 6 = e + 7 ).

(7) γ P 7 (1) = ∅.

(since e + 7 + e - 7 = 0 ). Then ,

(2) If m = 2, then J 2 = {1} and c

(3) If m = 3, then J 3 = {1} and c

(4) If m = 4, then J 4 = {1} and c

Recall the definition of d i for D,

c j a γ j .

Then we have

Hence in (7.1)

and we conclude that -(K Xr + D) is ample if and only if k i < 0 for all 1 ≤ i ≤ r. This completes the proof of the theorem.

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 R and let K be a (closed) cone in

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 N E(X) ⊂ N 1 (X) is called Mori if R • K X < 0, where K X is the canonical divisor in X. Recall that N E(X r ) is a strongly convex rational polyhedral cone of maximal dimension in N 1 (X r ). We prove, Theorem 8.1.

(1) The class of curves r(P i ) for 1 ≤ i ≤ r are all extremal rays in the Mori cone N E(X r ) of X r .

(2) Fix 1 ≤ i ≤ r, the class of curve r(P i ) is Mori ray if and only if either |γ P i (1)| = 0, or |γ P i (1)| = 1 with c j = 1 for γ j ∈ γ 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 ≤ i ≤ r are all extremal rays in N E(X r ). Hence for 1 ≤ i ≤ r, r(P i ) is Mori if K Xr • r(P i ) < 0. Since K Xr = -ρ∈Σ(1) D ρ , we can see by Corollary 4.13 and by similar arguments as in the proof of Lemma 5.1, K Xr • r(P i ) = -2 + γ j ∈γ P i (1) c j .

(8.1)

Thus if K Xr • r(P i ) < 0, then γ j ∈γ P i (1) c j < 2.

As c j are all positive integers (see (4.4)), we get either |γ P i (1)| = 0, or |γ P i (1)| = 1 and c j = 1 for γ j ∈ γ P i (1). Similarly, by using (8.1) we can prove the converse. This completes the proof of the theorem.