B Narasimha Chary

A NOTE ON TORIC DEGENERATION OF A BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY

Keywords: Bott-Samelson-Demazure-Hansen varieties, canonical line bundle, tangent bundle and toric varieties

. We give some applications to BSDH varieties. Precisely, we classify Fano, weak Fano and log Fano BSDH varieties and their toric limits in Kac-Moody setting. We prove some vanishing theorems for the cohomology of tangent bundle (and line bundles) on BSDH varieties. We also recover the results in [PK16], by toric methods.

Introduction

Bott-Samelson-Demazure-Hansen (for short, BSDH) varieties are natural desingularizations of Schubert varieties in the flag varieties. These were algebraically constructed by M. Demazure and H.C. Hansen independently by adapting a differential geometric approach from the paper of Bott and Samelson (see [START_REF] Bott | Applications of the theory of Morse to symmetric spaces[END_REF], [START_REF] Demazure | Désingularisation des variétés de Schubert généralisées[END_REF] and [START_REF] Hansen | On cycles in flag manifolds[END_REF]). Briefly, the BSDH varieties are iterated projective line bundles, given by factoring the Schubert variety using Bruhat decomposition. These varieties depend on the given expression of the Weyl group element corresponding to the Schubert variety (see for instance [START_REF]Automorphism group of a Bott-Samelson-Demazure-Hansen variety[END_REF]Page 32]). We also see in this paper some properties of these varieties which depend on the given expression.

In [START_REF] Grossberg | Bott towers, complete integrability, and the extended character of representations[END_REF], M. Grossberg and Y. Karshon constructed toric degenerations of BSDH varieties by complex geometric methods. In [START_REF] Pasquier | Vanishing theorem for the cohomology of line bundles on Bott-Samelson varieties[END_REF] B. Pasquier and in [START_REF] Parameswaran | Toric degeneration of Bott-Samelson-Demazure-Hansen varieties[END_REF] A.J. Parameswaran and P. Karuppuchamy constructed these toric degenerations algebraically. B. Pasquier used these degenerations to study the cohomology of line bundles on BSDH varieties (see [START_REF] Pasquier | Vanishing theorem for the cohomology of line bundles on Bott-Samelson varieties[END_REF]). In [START_REF] Parameswaran | Toric degeneration of Bott-Samelson-Demazure-Hansen varieties[END_REF], the authors studied the limiting toric variety for a simple simply connected algebraic group by geometric methods. In this paper we study the limiting toric variety of a BSDH variety in more detail by methods of toric geometry and we prove some applications to BSDH varieties. We also recover the results in [START_REF] Parameswaran | Toric degeneration of Bott-Samelson-Demazure-Hansen varieties[END_REF] and extend them to the Kac-Moody setting. The key idea for many results in this article is that the toric limit is a 'Bott tower'. These are studied in [Cha] and some of their properties can be transferred to BSDH varieties by using the semi-continuity theorem.

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

Let G be a Kac-Moody group over the field of complex numbers (for the definition see [START_REF] Kumar | Kac-Moody groups, their flag varieties and representation theory[END_REF]). Let B be a Borel subgroup containing a fixed maximal torus T . Let W be the Weyl group corresponding to the pair (G, B, T ) and let w ∈ W . Let w := s β 1 • • • s βn be an expression (possibly non-reduced) of w in simple reflections and let Z( w) be the BSDH variety corresponding to w (see Section 2). Let Y w be the toric limit of Z( w) constructed as in [START_REF] Pasquier | Vanishing theorem for the cohomology of line bundles on Bott-Samelson varieties[END_REF] and [START_REF] Parameswaran | Toric degeneration of Bott-Samelson-Demazure-Hansen varieties[END_REF] (see Section 3). We see that Y w is a Bott tower, the iterated P 1 -bundle over a point {pt} where each P 1 -bundle is the projectivization of a rank 2 decomposable vector bundle (see Corollary 4.4). We prove that the ample cone Amp(Y w) of Y w can be identified with a subcone of the ample cone Amp(Z( w)) of Z( w) (see Corollary 5.1).

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 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.

When G is a simple algebraic group and the expression w is reduced, Fanoness and weak Fanoness of the BSDH variety Z( w) are considered in [START_REF] Bott-Samelson-Demazure-Hansen Varieties | On Fano and weak[END_REF]. Here we have the following results in Kac-Moody setting. Let w = s

β 1 • • • s β i • • • s β j • • •
s βr be an expression (remember that β k 's are simple roots). Let β ij := β j , βi , where βi is the co-root of β i . Now we define some conditions on the expression w (see Section 6 for examples and also see [Cha, Section 1]). 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 }). • N 1 i is the condition that (i) |η + i | = 0, |η - i | ≤ 1, 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 - 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 the expression w 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. Theorem (See Lemma 6.1 and Theorem 6.3).

(1) If w satisfies I, then Y w and Z( w) are Fano.

(2) If w satisfies II, then Y w and Z( w) are weak Fano.

In [START_REF]Automorphism group of a Bott-Samelson-Demazure-Hansen variety[END_REF] and [START_REF] Chary | Rigidity of a Bott-Samelson-Demazure-Hansen variety for P Sp(2n, C)[END_REF], we have obtained some vanishing results for the cohomology of tangent bundle of Z( w), when G is finite dimensional and w is reduced (see [CKP15, Section 3] and [CK17, Theorem 8.1] ). The case w is non-reduced is considered in [CKP].

Here we get some vanishing results in Kac-Moody setting. Let T Z( w) denote the tangent bundle of Z( w).

Corollary (see Corollary 6.6). If w satisfies I, then H i (Z( w), T Z( w) ) = 0 for all i ≥ 1. In particular, Z( w) is locally rigid.

In [START_REF] Anderson | Schubert varieties are log Fano over the integers[END_REF], D. Anderson and A. Stapledon studied the log Fanoness of Schubert varieties, and in [START_REF] Anderson | Effective divisors on Bott-Samelson varieties[END_REF], log Fanoness of BSDH varieties is studied for chosen divisors. Let D be a divisor in Z( w) with support in the boundary of Z( w). For 1 ≤ i ≤ r, we define some constants f i which again depend on the given expression w (for more details see Section 6).

Corollary (see Corollary 6.7). The pair

(Z( w), D) is log Fano if f i > 0 for all 1 ≤ i ≤ r.
The article is organized as follows: In section 2, we recall the construction of BSDH varieties. In Section 3, we give the algebraic construction of toric degeneration of a BSDH variety. In Section 4, we describe the limiting toric variety as an iterated P 1 -bundle. In Section 5, we see some vanishing results of cohomology of line bundles on BSDH varieties. Section 6 contains the results on Fano, weak Fano and log Fano properties of BSDH varieties and their toric limits. We also study the vanishing results on cohomology of tangent bundle on BSDH varieties. In Section 7, we recover the results in [START_REF] Parameswaran | Toric degeneration of Bott-Samelson-Demazure-Hansen varieties[END_REF] by toric methods.

Preliminaries

In this section we recall the construction of Bott-Samelson-Demazure-Hansen varieties (see [START_REF] Brion | Frobenius splitting methods in geometry and representation theory[END_REF] and [START_REF] Kumar | Kac-Moody groups, their flag varieties and representation theory[END_REF]) and we recall some definitions in toric geometry which are used in this article (for more details on toric varieties see [START_REF] Cox | Toric varieties[END_REF] and also [START_REF] Fulton | Introduction to toric varieties[END_REF]). We work over the field of complex numbers throughout.

BSDH varieties.

Let A = (a ij ) 1≤i,j≤n be a generalized Cartan matrix. Let G be the Kac-Moody group associated to A (see [START_REF] Kumar | Kac-Moody groups, their flag varieties and representation theory[END_REF]Chapter IV]). Fix a maximal torus T and a Borel subgroup B containing T . Let S := {α 1 , . . . , α n } be the set of all simple roots of (G, B, T ). We denote s α i the simple reflection corresponding to α i . Note that the Weyl group W of G is generated by

{s α i : 1 ≤ i ≤ n}.
Let w ∈ W , an expression w of w is a sequence (s β 1 , . . . , s βr ) of simple reflections s β 1 , . . . , s βr such that w = s β 1 • • • s βr . An expression w of w is said to be reduced if the number r of simple reflections is minimal. In such case we call r the length of w. By abuse of notation, we also denote the expression w by w = s β 1 • • • s βr . For α ∈ S, we denote P α , the minimal parabolic subgroup of G generated by B and a representative of s α . Definition 2.1. Let w ∈ W and w := s β 1 • • • s βr be an expression (not necessarily reduced) of w. The Bott-Samelson-Demazure-Hansen (for short, BSDH) variety corresponding to w is Z( w) := P β 1 × • • • × P βr /B r , where the action of B r on P β 1 × • • • × P βr is defined by

(p 1 , . . . , p r ) • (b 1 , . . . , b r ) = (p 1 b 1 , b -1 1 p 2 b 2 , . . . , b -1 r-1 p r b r ) for all p i ∈ P β i , b i ∈ B.
These are smooth projective varieties of dimension r. There is a natural morphism φ w : Z( w) -→ G/B defined by

[(p 1 , . . . , p r )] → p 1 • • • p r B.
If w is reduced, the BSDH variety Z( w) is a natural desingularization of the Schubert variety, the B-orbit closure of wB/B in G/B (see [START_REF] Demazure | Désingularisation des variétés de Schubert généralisées[END_REF], [START_REF] Hansen | On cycles in flag manifolds[END_REF] and [Kum12, Chapter VIII]). We can also construct the BSDH variety as an iterated P 1 -bundles. Let w := s β 1 • • • s β r-1 . Let f : G/B -→ G/P βr be the map given by gB → gP βr and let p : Z( w ) -→ G/P βr be the map given by [(p 1 , . . . , p r-1 )] → p 1 • • • p r-1 P βr . Then we have the following cartesian diagram (see [BK07, Page 66] and [Kum12, Chapter VII]):

Z( w) = Z( w ) × G/P βr G/B φ w / / f w G/B f Z( w ) p / / G/P βr
Note that f w is a P 1 -fibration and the relative tangent bundle T f w of f w is φ * w(L βr ), where L βr is the homogeneous line bundle on G/B corresponding to β r . Using the cohomology of the relative tangent bundle T f w we studied the cohomology of the tangent bundle of Z( w), when G is finite dimensional and w is a reduced expression (see [START_REF]Automorphism group of a Bott-Samelson-Demazure-Hansen variety[END_REF] and [START_REF] Chary | Rigidity of a Bott-Samelson-Demazure-Hansen variety for P Sp(2n, C)[END_REF]). The fibration f w comes with a natural section σ w : Z( w ) → Z( w) induced by the projection

P β 1 × • • • × P βr → P β 1 × • • • × P β r-1 .
For the toric limits we get two natural sections, as will be explained in Section 3. For all i ∈ {1, . . . , r}, we denote Z i , the divisor in Z( w) defined by

{[(p 1 , . . . , p r )] ∈ Z( w) : p i ∈ B}.
In [START_REF] Lauritzen | Line bundles on Bott-Samelson varieties[END_REF], N. Lauritzen and J.F. Thomsen proved that Z i s forms a basis of the Picard group of Z( w) and they also proved that if w is a reduced expression these form a basis of the monoid of effective divisors (see [LT04, Proposition 3.5]). Recently, the effective divisors of Z( w) for w non-reduced case have been considered in [START_REF] Anderson | Effective divisors on Bott-Samelson varieties[END_REF].

The BSDH variety can be described also as an iterated projective line bundle, where each projective bundle is the projectivization of certain rank 2 vector bundle (not necessarily decomposable). In Section 4, we see the toric degeneration of a BSDH variety (constructed in Section 3) is a Bott tower, the iterated P 1 -bundle over a point {pt}, where each P 1 -bundle is the projectivization of a rank 2 decomposable vector bundle.

Toric varieties.

Definition 2.2. 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 denote the fan corresponding to a toric variety by Σ and the collection of cones of dimension s in Σ by Σ(s) for 1 ≤ s ≤ n. For each cone σ ∈ Σ, we denote V (σ), the orbit closure of the orbit corresponding to cone σ. For each σ ∈ Σ, σ(1) := σ ∩ Σ(1). For each ρ ∈ Σ(1), we can associate a divisor in X, we denote it by D ρ (see [START_REF] Cox | Toric varieties[END_REF]Chapter 4] for more details). We recall the following:

Definition 2.3.
(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.

(2) Let P = {ρ 1 , . . . , ρ k } be a primitive collection in a complete simplicial fan Σ.

Recall u ρ be 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 ) = 0. (2.1)
where c ρ ∈ Q >0 . Then we call (2.1) the primitive relation of X corresponding to P.

(3) For a primitive relation P , we can associate an element r(P ) in N 1 (X), where N 1 (X) is the real vector space of numerical classes of one-cycles in X (see [START_REF] Cox | Toric varieties[END_REF] Page 305]).

Toric degeneration of a BSDH variety

In [START_REF] Grossberg | Bott towers, complete integrability, and the extended character of representations[END_REF], toric degenerations of BSDH varieties were constructed by complex geometric methods. In [START_REF] Pasquier | Vanishing theorem for the cohomology of line bundles on Bott-Samelson varieties[END_REF] and [START_REF] Parameswaran | Toric degeneration of Bott-Samelson-Demazure-Hansen varieties[END_REF] they have given an algebraic construction for toric degeneration of a BSDH variety. We recall the algebraic construction here.

Note that the simple roots are linearly independent elements in the character group of G. Let N be the lattice of one-parameter subgroups of T . We can choose a positive integer q and an injective morphism λ : G m -→ T (i.e. λ ∈ N and λ is injective) such that for all 1 ≤ i ≤ n and u ∈ G m , α i (λ(u)) = u q (see [Pas10, Page 2836]). When G is finite dimensional, for each one-parameter subgroup λ ∈ N , define

P (λ) := {g ∈ G : lim u→0 λ(u)gλ(u) -1 exists in G}.
The set P (λ) is a parabolic subgroup and the unipotent radical R u (P (λ)) of P (λ) is given by R u (P (λ)) = {g ∈ G : lim u→0 λ(u)gλ(u) -1 is identity in G}.

Any parabolic subgroup of G is of this form (see [Spr10, Proposition 8.4.5]). Choose a one-parameter subgroup λ ∈ N such that the corresponding parabolic subgroup is B. Let us define an endomorphism of G for all u ∈ G m by Ψu : G → G, g → λ(u)gλ(u) -1 .

Let B be the set of all endomorphisms of B. Now define a morphism

Ψ : G m → B by u → Ψu | B .
This map can be extended to 0 and for all x ∈ U , Ψ u | B (x) goes to identity when u goes to zero. Let A 1 := SpecC[t] be the affine line over C. We denote for all u ∈ A 1 , Ψ u the image of u in B. Note that Ψ u is the identity on T and Ψ 0 is the projection from B to T . Let w = s β 1 • • • s βr be an expression.

Definition 3.1.

(i) Let X be the variety defined by

X := A 1 × P β 1 × • • • × P βr /B r ,
where the action of B r on

A 1 × P β 1 × • • • × P βr is given by (u, p 1 , . . . , p r ) • (b 1 , . . . , b r ) = (u, p 1 b 1 , Ψ u (b 1 ) -1 p 2 b 2 , . . . , Ψ u (b r-1 ) -1 p r b r ).
(ii) For all i ∈ {1, . . . , r}, we denote Z i the divisor in X defined by

{(u, p 1 , . . . , p r ) ∈ Z : p i ∈ B}.
Note that X and Z i s are integral. Let π : X → A 1 be the projection onto the first factor. Then we have the following theorem (see [Pas10, Proposition 1.3 and 1.4] and [PK16, Theorem 9] ).

Theorem 3.2.

(1) π : X → A 1 is a smooth projective morphism.

(2) For all u ∈ A 1 \ {0}, the fiber π -1 (u) is isomorphic to the BSDH variety Z( w) such that π -1 (u) ∩ Z i corresponds to the divisor Z i in Z( w). (3) π -1 (0) is a smooth projective toric variety.

We denote X u := π -1 (u) for u ∈ A 1 and the limiting toric variety X 0 = π -1 (0) by Y w.

Connection to Bott towers

In this section we describe the toric limit Y w as an iterated P 1 -bundle. We also recall some results on Bott towers from [Cha]. Let {e + 1 , . . . , e + r } be the standard basis of the lattice Z r . Define for all i ∈ {1, . . . , r},

e - i := -e + i - j>i β ij e + j , (4.1)
where β ij := β j , βi . The following proposition will give the description of the fan of the toric variety Y w (see [Pas10, Proposition 1.4]).

Proposition 4.1.

(1) The fan Σ of the smooth toric variety Y w consists of the cones generated by subsets of {e + 1 , . . . , e + r , e - 1 , . . . , e - r } and containing no subset of the form {e + i , e - i }.

(2) For all i ∈ {1, . . . , r}, Z 0 i is the irreducible (C * ) r -stable divisor in Y w corresponding to the one-dimensional cone of Σ generated by e + i and these form a basis of the divisor class group of Y w.

Note that the maximal cones of Σ are generated by {e i : 1 ≤ i ≤ r, ∈ {+, -}} . We denote the divisor corresponding to the one-dimensional cone ρ i generated by e i by D ρ i for ∈ {+, -}. Let w := s β 1 • • • s β r-1 . Then we get a toric morphism f r : Y w → Y w induced by the lattice map f r : Z r → Z r-1 , the projection onto the first r -1 coordinates. We prove, Lemma 4.2.

(1) f r : Y w → Y w is a toric P 1 -fibration with two disjoint toric sections.

(2) Y w P(O Y w ⊕ L ) for some unique line bundle L on Y w .

Proof. Let Σ be the fan corresponding to the toric variety Y w . From the above proposition, we can see that Σ has a splitting by Σ and {e + r , 0, e - r }. Then by [CLS11, Theorem 3.3.19],

f r : Y w → Y w is a locally trivial fibration with the fan Σ F of the fiber being {e + r , 0, e - r }. Since Σ F is the fan of the projective line P 1 , we conclude f r is a toric P 1 -fibration. As toric sections of the toric fibration correspond to the maximal cones in Σ F , we get two disjoint toric sections for f r . This proves (1).

Proof of (2): Since f r : Y w → Y w is P 1 -fibration with a section, we see Y w is a projective bundle P(E ) over Y w corresponding to a rank 2 vector bundle E on Y w (see for example [Har77, Chapter V, Proposition 2.2, page 370]).

Recall that the sections of projective bundle Y w = P(E ) correspond to the quotient line bundles of E (see [START_REF] Hartshorne | Algebraic geometry[END_REF]

-→ Y r-1 π r-1 -→ • • • π 2 -→ Y 1 = P 1 π 1 -→ Y 0 = {pt},
where

Y i = P(O Y i-1 ⊕ L i-1
) for a line bundle L i-1 over Y i-1 for all 1 ≤ i ≤ r and P(-) denotes the projectivization (see for more detalis [START_REF] Civan | Bott towers, crosspolytopes and torus actions[END_REF] and also [Cha, Section 2]).

Then by definition of Bott tower and by Lemma 4.2(2) we get:

Corollary 4.4. The toric limit Y w is a Bott tower.

We have the following situation:

P 1 P 1 Z( w) Y w Z( w ) Y w f w fr
Recall that the Bott towers bijectively correspond to the upper triangular matrices with integer entries (see [Civ05, Section 3]). Here the upper triangular matrix M w corresponding to Y w is given by

M w =       1 β 12 β 13 . . . β
i := {ρ + i , ρ - i } for 1 ≤ i ≤ r.
Then by [Cha, Lemma 4.3], {P i : 1 ≤ i ≤ r} is the set of all primitive collections of Y w. For each 1 ≤ i ≤ r, we denote the cone in the definition of primitive relation (see Section 2) corresponding to P i by γ P i . Let D = ρ∈Σ(1) a ρ D ρ be a toric divisor in Y w with a ρ ∈ Z 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 recall the following from [Cha, Lemma 5.1]:

Lemma 4.5.

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

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

Also note that the conditions I and II on w are same as the conditions on M w as in [Cha].

Vanishing results on Cohomology of certain line bundles on BSDH varieties

Let X be a smooth projective variety. Recall N 1 (X) denote the real finite dimensional vector space of numerical classes of real divisors in X (see [Kle66, §1, Chapter IV]). The ample cone Amp(X) of X is the cone in N 1 (X) generated by classes of ample divisors. 5.1. Ample cone of the toric limit of BSDH variety. In [START_REF] Lauritzen | Line bundles on Bott-Samelson varieties[END_REF], the ampleness of line bundles on BSDH variety Z( w) is studied. Now we compare the ample cone of the toric limit Y w with that of the BSDH-variety Z( w) as a consequence of Theorem 3.2.

Corollary 5.1. The ample cone Amp(Y w) of Y w can be identified with a subcone of the ample cone Amp(Z( w)) of Z( w).

Proof. By Theorem 3.2, π : X → A 1 is a smooth projective morphism with X 0 = Y w and X u = Z( w) for u = 0. Let L = {L u : u ∈ A 1 } be a line bundle on π : X → A 1 with L 0 is an ample line bundle on Y w. Note that the ampleness of line bundle is an open condition for the proper morphism π, i.e. there exists an open subset U in A 1 containing 0 such that L u is an ample line bundle on X u for all u ∈ U (see [Laz04, Theorem 1.2.17]). Hence we can identity Amp(Y w) with a subcone of Amp(Z( w)).

Vanishing results.

In [START_REF] Pasquier | Vanishing theorem for the cohomology of line bundles on Bott-Samelson varieties[END_REF], B. Pasquier obtained vanishing theorems for the cohomology of certain line bundles on BSDH varieties, by using combinatorics of the toric limit (see [Pas10, Theorem 0.1]). Here we see some vanishing results for the cohomology of certain line bundles on BSDH varieties. 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. Let ∈ {+, -}. Define Σ(1) := {ρ i : 1 ≤ i ≤ r}.
Then we can write a toric divisor D in Y w as follows:

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 .
Recall d i from Section 4,

d i := (a ρ + i + a ρ - i - γ j ∈γ P i (1) c j a γ j ),
Let D = r i=1 g i Z i be a divisor in Z( w), where Z i is as in Section 2 for 1 ≤ i ≤ r. Lemma 5.2. If d i ≥ 0 for all 1 ≤ i ≤ r, then H j (Z( w), D ) = 0 for all j > 0.

Proof. If d i ≥ 0 for all 1 ≤ i ≤ r, by Lemma 4.5, ρ∈Σ(1) a ρ D ρ is a nef divisor in Y w. Then we have

H j (Y w, ρ∈Σ (1)
a ρ D ρ ) = 0 for all j > 0 (5.1)

(see [CLS11, Theorem 9.2.3, page 410] or [Oda88, Theorem 2.7, page 77]). Recall that by Theorem 3.2, we have

Z x i = Z i for 0 = x ∈ k and Z 0 i = D ρ + i . By [Cha, Corollary 3.3], we can write D = ρ∈Σ(1) a ρ D ρ ∼ r i=1 g i D ρ + i .
Hence by (5.1), Theorem 3.2 and by semi-continuity theorem (see [START_REF] Hartshorne | Algebraic geometry[END_REF]Theorem 12.8]), we get H j (Z( w), D ) = 0 for all j > 0.

6. Fano, Weak Fano and log Fano BSDH varities 6.1. Fano and weak Fano properties. In this section, we observe that Fano and weak Fano properties for BSDH variety Z( w) depend on the given expression w. We use the terminology from 1. First we discuss the conditions I and II with some examples. We use the ordering of simple roots as in [Hum72, Page 58].

The condition I:

(1) Special case: |η + i | = 0 and |η - i | = 0. This condition means that the expression w is fully commutative without repeating the simple reflections. For example if G = SL(n, C) and w = s α 1 s α 3 • • • s αr , 1 < r ≤ n -1 and r is odd, then |η + i | = 0 and |η - i | = 0 for all i. Hence w satisfies the condition I and also observe that in this case we have

Y w Z( w) P 1 × • • • × P 1 (dim(Z( w)) times ).
(2) Let G = SL(n, C) and fix 1 ≤ j < r ≤ n -1 such that j is even and r is odd.

Let w = s α 1 s α 3 • • • s α j-3 s α j-1 s α j s α j+1 s α j+3 • • • s αr . Note that s α j appears

The condition II:

Again, let G = SL(n, C) and fix 1 ≤ j < r ≤ n -1 such that j is even and r is odd. Let w = s α 1 s α 3 • • • s α j-3 s α j s α j-1 s α j+1 s α j+3 • • • s αr (observe that we interchanged s α j and s α j-1 in the example of condition II). Then |η + i | = 0 and |η - i | ≤ 2 for all i. Let p be the 'position of s α j ' in the expression w, then |η - i | = 0 for all i = p and |η - p | = 2 with β pp+1 = -1 = β pp+1 . Hence w satisfies the condition II but not I.

Let w = s α 1 s α 3 s α 1 . Then |η + 1 | = 1 with β 13 = 2, and |η - 1 | = |η + 2 | = η - 2 | = 0 . Hence w1 satisfies II but not I.
Observe that the condition |η - i | = 1 and β il = -2, happens only in non-simply laced cases. Let G = SO(5, k) (i.e. G is of type B 2 ), let w1 = s α 2 s α 1 and w2 = s α 1 s α 2 . Recall that we have α 1 , α 2 = -2 and α 2 , α 1 = -1. Then Hence w1 satisfies II but not I and w2 satisfies I.

Let G be of type G 2 (with α 1 , α 2 = -1 and α 2 , α 1 = -3). Let w1 = s α 2 s α 1 and w2 = s α 1 s α 2 . Then Hence w1 satisfies I and w2 does not satisfy any of the conditions I or II.

Now we have the following result:

Lemma 6.1.

(1) Y w is Fano if and only if w satisfies I.

(2) Y w is weak Fano if and only if w satisfies II.

Proof. This follows from Corollary 4.2 and [Cha, Theorem 6.3].

Recall the following (see for instance [START_REF] Chary | On Mori cone of Bott towers[END_REF]Corollary 6.2]): Lemma 6.2. Let X be a smooth projective variety and D be an effective divisor. Let supp(D) denote the support of D. If X \ supp(D) is affine, then D is big.

We prove the following: Theorem 6.3.

(1) If w satisfies I, then Z( w) is Fano.

(2) If w satisfies II, then Z( w) is weak Fano.

Proof. First recall that the canonical line bundle O Z( w) (K Z( w) ) of Z( w) is given by

O Z( w) (K Z( w) ) = O Z( w) (-∂Z( w)) ⊗ L(-δ),
where ∂Z( w) is the boundary divisor of Z( w) and δ ∈ N such that δ, α = 1 for all α ∈ S, where α is the co-root of α (see [Kum12, Proposition 8.1.2] and also [Ram85, Proposition 2]). Note that if G is finite dimensional, δ is half sum of the positive roots.

By Theorem 3.2, φ : X → A 1 is a smooth projective morphism with X 0 = Y w and X u = Z( w) for 0 = u ∈ A 1 .
Proof of (1): By [Laz04, Theorem 1.2.17], if -K X 0 is ample then -K Xu is ample for u = 0. By Lemma 6.1, -K Y w is ample if and only if w satisfies I. Hence we conclude that if w satisfies I, then Z( w) is Fano.

Proof of (2): First we prove -K Z( w) is big. Let

Z 0 := Z( w) \ ∂Z( w).
Note that Z 0 is an open affine subset of Z( w). Then by Lemma 6.2, ∂Z( w) is big. Since

O(-K Z( w) ) = O(∂Z( w)) ⊗ L(δ)
and L(δ) is nef, we conclude -K Z( w) is big, as tensor product of a big and a nef line bundles is again a big line bundle. By [Laz04, Theorem 1.4.14] and X u = Z( w) for u = 0, we can see that if -K X 0 is nef then -K Xu is also nef for u = 0. Therefore, (2) follows from Lemma 6.1(2).

There exists expressions w such that the BSDH variety Z( w) is Fano (respectively, weak Fano) but the toric limit Y w is not Fano (respectively, not weak Fano).

Example 6.4. Let G = SL(4, C).

(1) Let w = s α 1 s α 1 . Then Z( w) P 1 × P 1 , which is Fano. The toric limit Y w P(O P 1 ⊕ O P 1 (2)). Since w does not satisfy I, then by Lemma 6.1, Y w is not Fano.

(2) Let w = s α 1 s α 2 s α 1 . Then it can be seen Z( w) is Fano (see [Cha17, Example 5.4]).

By Lemma 6.1, the toric limit Y w is weak Fano but not Fano.

Example 6.5. Let G = SO(5, k), i.e. G is of type B 2 . Let w = s α 1 s α 2 s α 1 . By Lemma 6.1, the toric limit Y w is not weak Fano. Also we can see Z( w) is weak Fano but not Fano (see [Cha17, Theorem 5.3]).

Local rigidity of BSDH varieties.

In this section we obtain some vanishing results for the cohomology of tangent bundle of the toric limit Y w and Z( w). Let T X denote the tangent bundle of X, where X = Y w or Z( w). Then we have Corollary 6.6.

(1) If w satisfies I, then H i (Y w, T Y w ) = 0 for all i ≥ 1. In particular, Y w is locally rigid.

(2) If w satisfies I, then H i (Z( w), T Z( w) ) = 0 for all i ≥ 1. In particular, Z( w) is locally rigid.

Proof. Proof of (1): If w satisfies I, then by Lemma 6.1, Y w is a Fano variety. By [BB96, Proposition 4.2], since Y w is a smooth Fano toric variety, we get H i (Y w, T Y w ) = 0 for all i ≥ 1.

Proof of (2): From Theorem 3.2, π : X → A 1 is a smooth projective morphism with X 0 = Y w and X u = Z( w) for u ∈ A 1 , u = 0. Hence (2) follows from (1) by semi-continuity theorem (see [START_REF] Hartshorne | Algebraic geometry[END_REF]Theorem 12.8]). 6.3. Log Fano BSDH varieties. In [START_REF] Anderson | Effective divisors on Bott-Samelson varieties[END_REF] and [START_REF] Anderson | Schubert varieties are log Fano over the integers[END_REF] log Fanoness of Schubert varieties and BSDH varieties were studied respectively. Now we characterize the (suitably chosen) Q-divisors D in Z( w)) for which (Z( w), D ) is log Fano. Recall that

Z i = {[(p 1 , . . . , p r )] ∈ Z( w) : p i ∈ B} is a divisor in Z( w) (see Section 2). Let γ i = s βr • • • s β i+1 (β i ) for 1 ≤ i ≤ r. Then, L(δ) = r i=1 b i Z i with b i = δ, γi = ht(γ i ), (6.1)
where δ is as in Section 6.1 (see page 10), L(δ) is the homogeneous line bundle on Z( w) corresponding to δ and ht(β) for a root β = n i=1 n i α i , is the height defined by ht(β) = n i=1 n i (see [MR85, Proof of Proposition 10]). When w is reduced, γ i is a positive root and we can see the relation (6.1) from the Chevalley formula for intersection of Schubert variety by a divisor (see [START_REF] Anderson | Schubert varieties are log Fano over the integers[END_REF]Page 410] or [START_REF] Chevalley | Sur les décompositions cellulaires des espaces G/B, Algebraic Groups and their Generalizations: Classical Methods[END_REF]). It is known that

-K Z( w) = r i=1 (b i + 1)Z i (6.2)
(see [START_REF] Mehta | Frobenius splitting and cohomology vanishing for Schubert varieties[END_REF]Proposition 4]). Let D = r i=1 a i Z i be a effective Q-divisor in Z( w), with D = 0, where i a i Z i = i a i Z i , x is the greatest integer ≤ x. Then by (6.2), we get

-(K Z( w) + D ) = r i=1 (b i + 1 + a i )Z i .
For 1 ≤ i ≤ r, define

f i := (b i + 1 + a i ) - γ j ∈γ P i (1) + c j (b j + 1 + a j ),
where γ P i (1) + := γ P i (1) ∩ {ρ + l : 1 ≤ l ≤ r} and γ P i is the cone as in (2.1) for the toric limit Y w.

Recall that 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]).

We prove, Corollary 6.7. The pair (Z( w), D ) is log Fano if f i > 0 for all 1 ≤ i ≤ r.

Proof. By definition of D , the pair (Z( w), D ) is log Fano if and only if -(K Z( w) + D ) is ample. Now we prove -(K Z( w) + D ) is ample if f i > 0 for all 1 ≤ i ≤ r. Recall that D ρ + i is the divisor corresponding to ρ + i ∈ Σ(1) and Z x i = π -1 (x) ∩ Z i for x ∈ k (see Section 2 and Section 3). By Theorem 3.2, we have

Z x i = Z i for x = 0 and Z 0 i = D ρ + i . (6.3)
Assume that f i > 0 for all 1 ≤ i ≤ r. By (6.3) and by semicontinuity (see [Laz04, Theorem 1.2.7]) to prove (Z( w), D ) is log Fano it is enough to prove r i=1

(b i + 1 + a i )D ρ + i is ample .
By Lemma 4.5, we see that r i=1 (b i + 1 + a i )D ρ + i is ample if and only if

f i = ((b i + 1 + a i ) - γ j ∈γ P i (1) + c j (b j + 1 + a j )) > 0 for all 1 ≤ i ≤ r.
Hence we conclude that (Z( w), D ) is log Fano.

(the image of L j by the Schubert section in Y w) form a basis of N 1 (Y w). By definition of L I , we have σ 0 r-1 (L j ) = L j for 1 ≤ j ≤ r -1 and hence the result follows.

Let 1 ≤ j ≤ r. Let D := {e l l : 1 ≤ l ≤ r and l = + for all l} . Let D j := {e l l : 1 ≤ l ≤ r and l = + for all l = l; j = -} .

Lemma 7.3. Fix 1 ≤ j ≤ r. Then the Schubert line L j is given by

L j = V (τ j ) , with τ j = σ ∩ σ j ,
intersection of two maximal cones in Σ, where σ (respectively, σ j ) is generated by D (respectively, D j ).

Proof. Let us consider the expression wj = s β 1 • • • s β j for 1 ≤ j < r. Let Σ j be the fan of the toric variety Y wj . By Lemma 4.2,

f j : Y wj → Y wj-1
is a P 1 -fibration induced by f j : Z j → Z j-1 the projection onto the first j -1 factors. Also note that the Schubert point in Y wj-1 corresponds to the maximal cone generated by {e + l : 1 ≤ l ≤ j -1} and the fan of the fiber is given by {e + j , 0, e - j }. Let σ j (respectively, σ j ) be the cone generated by {e + l : 1 ≤ l ≤ j} (respectively, {e + l : 1 ≤ l ≤ j -1} ∪ {e - j } ). Then by definition of Schubert line L j , we can see that L j is the curve in Y wj given by L j = V (τ j ), where τ j ∈ Σ j and τ j = σ j ∩ σ j .

Since the Schubert section of f k for (j ≤ k ≤ r) corresponds to e + k , we see

σ 0 r • • • • • σ 0 j+1 (L j )
, by abuse of notation we also denote it again by L j in Y w, is given by

L j = V (τ j ) with τ = σ ∩ σ j ,
where σ and σ j are as described in the statement. This completes the proof of the lemma.

Let τ be a cone of dimension r -1 which is a wall, that is τ = σ ∩σ for some σ, σ ∈ Σ of dimension r. Let σ (respectively, σ ) be 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 (7.1)
The relation (7.1) called wall relation and we have

D ρ • V (τ ) =      b i if ρ = ρ i and i ∈ {2, 3, . . . ,
K Y w • L j = -2 - k>j β kj .
Proof. By definition of e - j , we have e + j + e - j + k>j β kj e + k = 0. (7.3) By Lemma 7.3, we have L j = V (τ ), with τ = σ ∩ σ where σ (respectively, σ ) is generated by {e l l : 1 ≤ l ≤ r, l = + for all l} (respectively, {e l l : l = + for 1 ≤ l ≤ r and l = j, j = -} ).

Hence (7.3) is the wall relation for the curve L j . Then by (7.2), we see that

D ρ • L j =      1 if ρ = ρ + j or ρ - j . β kj if ρ = ρ + k and k > j. 0 otherwise. Since K Y w = -ρ∈Σ(1) D ρ , we get K Y w • L j = -2 - k>j β kj .
This completes the proof of the proposition. Now onwards we denote the subsequence (i 1 , . . . , i m ) by

I i 1 . Let D i 1 := {e l l : 1 ≤ l ≤ r and l = + if l / ∈ I i 1 \ {i 1 } -if l ∈ I i 1 }. Let D i 1 := {e l l : 1 ≤ l ≤ r and l = + if l / ∈ I i 1 -if l ∈ I i 1 }.
Proposition 7.5. The curve L I i 1 is given by

L I i 1 = V (τ i 1 ) with τ i 1 = σ i 1 ∩ σ i 1 ,
where σ i 1 (respectively, σ i 1 ) is the cone generated by D i 1 (respectively, D i 1 ).

Proof. As in the proof of Lemma 7.3, we start with j = i 1 and L i 1 is the Schubert line in Y wi 1 . By Lemma 7.3, we have

L i 1 = V (τ i 1 ) with τ i 1 = σ i 1 ∩ σ i 1 .
By definition of L I , we have

σ 0 i 2 -1 • • • • • σ 0 i 1 +1 (L i 1 ) = L i 1 in Y wi 2 -1 and σ 1 i 2 • σ 0 i 2 -1 • • • • • σ 0 i 1 +1 (L i 1 ) = L {i 1 ,i 2 } in Y wi 2 .
By repeating the process we conclude that

L I i 1 = V (τ i 1 ) with τ i 1 = σ i 1 ∩ σ i 1 ,
where σ i 1 and σ i 1 are as described in the statement. This completes the proof of the proposition.

Recall N E(X) is 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 (see for instance [CLS11, Chapter 6, page 293]). Now we describe the Mori cone of the toric limit Y w in terms of the curves L I i j 's defined above. For this we need the following notation (see also [Cha]). Fix 1 ≤ i ≤ r. Define:

(1) Let r ≥ j > j 1 = i ≥ 1 and define a 1,j := β j 1 j .

(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) Let Ĩi := {i = j 1 , . . . , j m }.

Example 7.6. Let G = SL(5, C) and let w = s β 1 • • • s β 7 = s α 2 s α 1 s α 3 s α 1 s α 2 s α 1 s α 2 . Let i = 1. Then j 1 = 1 and (1) a 1,2 = β 12 = β 2 , β1 = α 1 , α2 = -1 ; (2) a 1,3 = β 13 = β 3 , β1 = α 3 , α2 = -1 ;

(3) a 1,4 = β 14 = β 4 , β1 = α 1 , α2 = -1 ; (4) a 1,5 = β 15 = β 5 , β1 = α 2 , α2 = 2 ;

(5) a 1,6 = β 16 = β 6 , β1 = α 1 , α2 = -1 ; (6) a 1,7 = β 17 = β 7 , β1 = α 2 , α2 = 2 .

Then by definition of j 2 , we have j 2 = 5 and (1) a 2,6 = β 15 β 56 -β 16 = β 5 , β1 β 6 , β5β 6 , β1 = α 1 , α2 = -1 ; (2) a 2,7 = β 15 β 57 -β 17 = α 2 , α2 = 2 . Then by definition of j 3 , we have j 3 = 6 and a 3,7 = -a 2,6 β 67 + a 2,7 = -( β 6 , β5 )( β 7 , β6 ) + ( β 7 , β5 ) = -(-1)(-1) + (2) = 1. Therefore, we get Ĩ1 = {1, 5, 6} . Lemma 7.11. Let X be a smooth projective toric variety of dimension r. Then X is Fano if and only if every extremal ray is Mori. (7.4) If X is Fano, then by definition, -K X is ample. By toric Kleiman criterion for ampleness [CLS11, Theorem 6.3.13], we can see that -K X • V (τ ) > 0 for all τ ∈ Σ(r -1). Then K X • V (τ ) < 0 for all τ ∈ Σ(r -1). In particular, every extremal ray is Mori.

Conversely, let R ≥0 [V (τ )] be an extremal ray, by assumption it is a Mori ray. Then by definition of a Mori ray, we have K X • V (τ ) < 0. This implies -K X • V (τ ) > 0. By (7.4), N E(X) is a polyhedral cone and hence the extremal rays generate the cone N E(X). Hence we see that -K X • C > 0 for all classes of curves [C] in N E(X). Again by toric Kleiman criterion for ampleness, we conclude that -K X is ample and hence X is Fano.

Then we have the following (see [PK16, Corollary 36]):

Corollary 7.12. The toric limit Y w is Fano if and only if every extremal ray in N E(Y w) is Mori.

  only once in the expression w and |η + i | = 0 for all i. Let p be the 'position of s α j ' in the expression w, then |η - i | = 0 for all i = p, p -1 and |η - p-1 | = 1 = |η - p | with β p-1p = -1 = β pp+1 . Hence w satisfies condition I.

  Proof. By [CLS11, Theorem 6.3.20] (Toric Cone Theorem), we haveN E(X) = τ ∈Σ(r-1) R ≥0 [V (τ )].

  Proposition 7.12]). Since Y w = P(E ) is projective line bundle on Y w with two disjoint sections, we see E is decomposable as a direct sum of line bundles on Y w . Definition 4.3. A Bott tower of height r is a sequence of projective bundles

	As for any line bundle L on Y w (see [Har77, Lemma 7.9]), we can assume without loss of P(E ) P(L ⊗ E ) generality E = O Y Y r πr

w ⊕ L for some unique line bundle L on Y w . Hence Y w P(O Y w ⊕ L ) and this completes the proof of the lemma.

  Let 1 ≤ j ≤ r and let L j be the Schubert line in Y w. Then,

		r}
	1	if ρ = ρ i and i ∈ {1, r + 1}	(7.2)
	0	otherwise
	(see [CLS11, Proposition 6.4.4 and eq. (6.4.6) page 303]). We prove the following (see
	[PK16, Proposition 33]):
	Proposition 7.4.

Acknowledgements: I would like to thank Michel Brion for valuable discussions and many critical comments.

More results on the toric limit

In this section we are going to recover the results of [START_REF] Parameswaran | Toric degeneration of Bott-Samelson-Demazure-Hansen varieties[END_REF] by using methods of toric geometry. In [START_REF] Parameswaran | Toric degeneration of Bott-Samelson-Demazure-Hansen varieties[END_REF], they have assumed that G is a simple algebraic group. In our situation G is a Kac-Moody group. Recall the following:

(1) w = s β 1 • • • s βr and w = s β 1 • • • s β r-1 .

(2) The toric morphism f r : Y w → Y w is induced by the lattice map f r : Z r → Z r-1 , the projection onto the first r -1 coordinates.

As we discussed in Section 3, there are two disjoint toric sections for the P 1 -fibration f r : Y w → Y w (see Lemma 4.2).

Definition 7.1.

(1) Schubert and non-Schubert sections: We call the section corresponding to the maximal cone ρ + r (respectively, ρ - r ) in Σ F (the fan of the fiber of f r ) by 'Schubert section σ 0 r-1 ' (respectively, 'non-Schubert section σ 1 r-1 ' ).

(2) Schubert point: Let σ ∈ Σ be the maximal cone generated by {e + 1 , . . . , e + r }. We call the point in Y w corresponding to the maximal cone σ by 'Schubert point'.

(3) Schubert line: We call the fiber of f r over the Schubert point by 'Schubert line L r '.

Note that these definitions agree with that of in [PK16, Section 4]. Now onwards we denote w = (1, . . . , r) (respectively, w = (1, . . . , r -1)) for the expression w = s

). Let I = (i 1 , . . . , i m ) be a subsequence of w. Inductively we define the curve L I corresponding to I. Let L I be the curve in Y w corresponding to the subsequence I = (i 1 , . . . , i m-1 ) of I . Then define

Recall some more notations. Let X be a smooth projective variety, we define

It is a well known fact that N 1 (X) is a finite dimensional real vector space dual to N 1 (X) (see [Kle66, Proposition 4, §1, Chapter IV]). We have the following result:

Lemma 7.2. The classes of Schubert lines L j , 1 ≤ j ≤ r form a basis of N 1 (Y w).

Proof. Proof is by induction on r. Assume that the result is true for r -1. Since Y w is a projective bundle over Y w (see Lemma 4.2), then by [Bar71, Lemma 1.1],

L r and σ 0 r-1 (L j ) for 1 ≤ j ≤ r Example 7.7. We use Example 7.6, for i = 1, we have I 1 = {1, 5, 6}. Then Fix 1 ≤ i ≤ r. Let I i := Ĩi = {i = j 1 , j 2 , . . . , j m } where j k 's are as above. With this notation we prove the following (see [START_REF] Parameswaran | Toric degeneration of Bott-Samelson-Demazure-Hansen varieties[END_REF]Theorem 22]):

Theorem 7.8. The set {L I i : 1 ≤ i ≤ r} of classes of curves forms a basis of N 1 (Y w) Z and every torus invariant curve in N 1 (Y w) lie in the cone generated by

Proof. By [Cha, Proposition 4.16], for 1 ≤ i ≤ r the curve r(P i ) (see Section 2 for the definition of r(P i )) is given by

, where τ i = σ i ∩ σ i and σ i (respectively, σ ) is generated by D i (respectively, D i ). From Proposition 7.5, we see that the class of the curve

Also by [START_REF] Chary | On Mori cone of Bott towers[END_REF]Corollary 4.8], the set {r(P i ) : 1 ≤ i ≤ r} forms a basis of N 1 (Y w) Z . Hence we conclude the assertion of the theorem.

We recall some definitions: Let V be a finite dimensional vector space over R and let K be a (closed) cone in

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. Then we have (see [START_REF] Parameswaran | Toric degeneration of Bott-Samelson-Demazure-Hansen varieties[END_REF]Theorem 30]):

Corollary 7.9. The extremal rays of the toric limit Y w are precisely the curves L I i for 1 ≤ i ≤ r.

Proof. This follows from the proof of the Theorem 7.8. Let X be a smooth projective variety. An extremal ray R in the Mori cone