Toric degeneration of Bott-Samelson-Demazure-Hansen varieties

Transcription

1 Toric degeneration of Bott-Samelson-Demazure-Hansen varieties Paramasamy Karuppuchamy and Parameswaran, A. J. Abstract In this paper we construct a degeneration of Bott-Samelson-Demazure-Hansen varieties to toric varieties in an algebraic family and study the geometry of the resulting toric varieties. We give a natural set of torus invariant curves that generate the Chow group of 1-cycles of the limiting toric variety and express the ample cone of this toric variety as a sub-cone of the ample cone of the corresponding Bott-Samelson-Demazure-Hansen variety. We also give a description of Extremal and Mori rays and determine when this toric variety is Fano. 1 Introduction Let G be an almost simple, simply connected, affine algebraic group defined over an algebraically closed field k of arbitrary characteristic. Let B be a Borel subgroup of G. Then G acts from the left on the flag variety G/B. The Paramasamy Karuppuchamy, Department of Mathematics and Statistics, Mail Stop 942,The University of Toledo 2801 W. Bancroft St.,Toledo, OH , USA. paramasamy@gmail.com School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai , India. param@math.tifr.res.in 1

2 B-invariant closed subvarieties of G/B are called Schubert varieties. Every Schubert variety is uniquely represented by an element of the Weyl group. After choosing a reduced expression for Weyl group elements as product of simple reflections, one constructs certain smooth birational modifications of the corresponding Schubert varieties. These desingularizations used to be called as Bott-Samelson varieties and the constructions were first described by Demezure and indepently by Hansen ([J], Chapetr 13, page 353). These constructions (cf. [D] or [J]) extend naturally to any given sequence of simple reflections. The resulting varieties are referred as Bott-Samelson-Demazure- Hansen varieties. We often abbreviate this long name and call these as BSDH varieties. For k = C, the field of complex numbers, the authors Grossberg and Karshon (cf. [GK]) obtained a family of complex structures on BSDH variety as differentiable manifold which degenerate to a different complex structure, with resulting manifold having a toric variety structure. We give an algebraic degeneration by constructing a smooth family of varieties parametrised by the affine line with general fibre isomorphic to the BSDH variety and the special fibre isomorphic to a smooth toric variety (cf. Section 3). In another direction it may be desirable to obtain degenerations of Schubert varieties using the degenerations of their BSDH resolutions. Using Standard Monomial basis Gonciulea and Lakshmibai [GL] degenerated Flag varieties and some Schubert varieties to toric varieties. Later Caldero [C] had constructed degeneration of Schubert varieties into toric varieties using Lusztig s canonical basis. Lauritzen and Thomsen have given an ampleness criterion for line bundles on these BSDH varieties (cf. [LT], Theorem 3.1). In fact they gave a set of line bundles and showed that the ample cone is the strict positive cone generated by these line bundles (cf. section 7). Here we describe the ample cone of the toric variety. We also give a necessary and sufficient condition 2

3 for the anti canonical bundle to be ample, i.e. whether this toric variety is a Fano variety. The paper is arranged as follows. In section 2 we fix some notations and recall some basic results. The main observation here is the relation between the self intersection number of a BSDH surface constructed using two simple reflections and the pairing between the corresponding roots (Lemma 3). Section 3 is devoted to obtain the degeneration of BSDH variety to a toric variety in an algebraic family (Theorem 9). In section 4 we study the Chow group of cycles of dimension 1 (curves) of the limiting toric variety. We label certain torus invariant curves and obtain some basic relation between them (Proposition 17, Remark 21). In section 5, we obtain a basis for the Chow group of one cycles of the limiting toric variety such that all torus invarian curves are non negative linear combination of elelments of this basis (Theorem 22). We also give two algorithms to find these torus invariant curves (Lemmas 23 and 24). In section 6 we study the Extremal rays and Mori rays and describe them completely in the limiting toric variety (Theorem 30, Theorem 35). The last section deals with description of the ample cone of these toric varieties as a subcone of the ample cone of the corresponding BSDH variety (Theorem 39). 2 Preliminaries Let G be an almost simple, simply connected, affine algebraic group defined over an algebraically closed field k of arbitrary characteristic. Fix a maximal torus T G. Then the Weyl group is defined as N(T )/T, where N(T ) is the normaliser of T in G. If we denote the character group of T by X(T ), then the Weyl group W has a faithful representation on the real vector space X(T ) R. Let (, ) be a non-degenerate W invariant pairing on X(T ) R. Let S, Φ +, Φ X(T ) be simple roots, positive roots and negative roots 3

4 respectively. Let U α be the root subgroup corresponding to the root α and B be the Borel subgroup of G generated by the root subgroups corresponding to the negative roots and maximal torus T. Let α := of α. 2α (α,α) be the co-root For a given simple root α S, let s α denote the simple reflection on X(T ) R defined by s α (x) = x (x, α )α. Then W is generated by {s α α S}. For a simple root α, the minimal parabolic subgroup P α is defined to be the subgroup generated by B and the root subgroup U α. The fundamental weight corresponding to the simple root α is denoted by ω α. Recall the following well-known description of parabolic subgroups of G. Let ξ : G m G be a one parameter subgroup of G. Define a G m -action on G by x g = ξ(x)gξ(x) 1. Then we have the following (cf. [S] p. 148): Lemma 1.. The set P (ξ) := {g G lim x 0 x g exists} is a parabolic subgroup and the unipotent radical R u (P (ξ)) of P (ξ) is given by {g G lim x 0 x g = identity}. Moreover any parabolic subgroup of G is of the form P (ξ) for some ξ : G m G. We fix a ξ : G m T once and for all such that B = P (ξ) and the unipotent radical B U of B is {g G lim x 0 x g = identity }. Then we have the following: Lemma 2.. Let A := spec k[t] denote the affine line over k. Define a family of homomorphisms φ x : B B parametrised by x A as follows: φ x (b) = ξ(x)bξ(x) 1 if x 0 and φ 0 (b) = lim x 0 ξ(x)bξ(x) 1 Then φ x is an automorphsm for x 0 and φ 0 is the natural Levi projection B T with unipotent radical B U of B as the kernel. Let w = (s 1, s 2,, s m ) be a sequence of simple reflections, where m is any positive integer. Each simple reflection s j is defined by a simple root, 4

5 say α j. We call m the length of the sequence and we denote it by l(w). Length of an empty sequence is defined to be 0. Note that this lenth is just the number of terms in the sequence. We also have a lenth function on the Weyl group W. Every element w of W can be written as a product of simple reflections, w = s j1 s j2 s jr. If w can not be written as of less than r number of simple reflections then this expression is called a reduced expression and r is called the the length of w. We have a natural map from (finite) sequences of simple relfections to the the Weyl group W. This maps a sequence w = (s 1, s 2,, s m ) to the element w = s 1 s 2 s m. But this representation may not be reduced and hence the length of the sequnce w may not be equal to the length of the corresponding Weyl group element w. A sequence of integers I = (i 1,, i r ) is called m-admissible if 1 i 1 < i 2 < < i r m. The entries i 1 and i r of I = (i 1,, i r ) are called initial and final entries respectively. Define the subsequence w I of w = (s 1, s 2,, s m ) for every m-admissible sequence I = (i 1,, i r ) by w I := (s i1,, s ir ). For 0 r m we define the truncated m-admissible sequences I[r] := (1, 2,, r) and [r]i := (r + 1, r + 2,, m), we denote the corresponding subsequence of simple reflections w I[r] and w [r]i by w[r] and [r]w respectively. Note that I[0] = [m]i is the empty sequnce of integers and w[m] = w = [0]w and [m]w = w[0] is the empty sequence of simple reflections. Let P w and B w denote the products P 1 P m and B B (m copies) respectively, where P j denotes the minimal parabolic subgroup P αj. The Bott-Samelson-Demazure-Hansen (BSDH) variety, Z w, is defined ( see Definition 5 and Definition 7 below for a twisted version) as the quotient Z w := P 1 P 2 P m B B B = P w B w 5

6 where the B w acts on P w from the right as follows: (p 1, p 2, p 3 p m )(b 1, b 2, b 3 b m ) = (p 1 b 1, b 1 1 p 2 b 2, b 1 2 p 3 b 3 b 1 m 1p m b m ) The BSDH variety also has the following inductive geometric construction. The induction is on the length of the sequence w = (s 1, s 2,, s m ). We construct the BSDH variety Z w[r] and a map f r : Z w[r] G/B inductively for all 0 r m. When l(w) = 0 i.e. w = w[0], we define the corresponding BSDH variety to be the unique B fixed point, the identity coset eb, of G/B. The map f 0 : Z w[0] G/B is the inclusion. When the l(w) = 1 i.e. w = w[1] = s 1, we define the corresponding BSDH variety Z w[1] = P1 /B as the fiber product Z w[0] B/P1 G/B Observe that our construction gives the BSDH variety Z w[1] with the map f 1 : Z w[1] G/B, the projection ψ 1 : Z w[1] Z w[0], and the section σ 0 : Z w[0] Z w[1]. Suppose the BSDH variety Z w[r] together with a map f r : Z w[r] G/B is already constructed. Now the fiber product Z w[r] B/Pr+1 G/B defines the BSDH variety Z w[r+1] and the map f r+1 and ψ r+1 are canonical projections. A section σ r to this projection ψ r+1 is equivalent to giving a lift of the map Z w[r] G/P r+1 to G/B. Our inductive procedure already provided us such a map, f r (see diagram below). f r+1 Z w[r+1] G/B σ r ψ r+1 π r+1 Z w[r] π r+1 f r G/Pr+1 In summary, we have indcutively constructed the following: 6

7 i) The BSDH variety Z w[r] with the map f r : Z w[r] G/B, for all r. ii) The projection ψ r : Z w[r] Z w[r 1], for 1 r m. iii) The section σ r : Z w[r] Z w[r+1], for 0 r m 1. The next two Lemmas appear in [[K], Lemma 3(a), Lemma 2(3)] in slightly different notations. Since these Lemmas are crucial for our computations we give a proof of one of these lemmas for the convenience of the reader. Lemma 3.. Let u = (s 1, s 2 ) be a sequence of simple reflections. Then the self intersection number of the section σ 1 (Z s1 ) in the surface Z u is (α 2, α1 ). By abuse of notation we sometimes denote this number (α 2, α1 ) by (2, 1). Proof: The self intersection number of the section σ 1 (Z s1 ) in the surface Z u is by definition the degree of the normal bundle N σ1 (Z s1 )/Z u. But the normal bundle of a section in a fibration can be identified with the restriction of the relative tangent bundle. Since this fibration is a fibre product of Z s1 G/P 2 with the natural fibration π 2 : G/B G/P 2, the relative tangent bundle is the pull back of the relative tangent bundle of π 2 Z s2 P 2 /B Z u f G/B σ 1 Z s1 ψ 2 π 2 G/P 2 Now the relative tangent bundle of π 2 is canonically identified with L α2, the line bundle on G/B associated to the character α 2. Hence it suffices to prove that the degree of L α2 restricted to f(σ 1 (Z s1 )) is (α 2, α1 ), as f defines an embedding. In fact f(σ 1 (Z s1 )) = P 1 /B G/B. 7

8 Using Part II, Proposition 5.2 of [J], we obtain the individual cohomology groups of the restriction of L λ for any character λ on P 1 /B, which in turn determine the line bundle. The Lemma will then follow by substituting the character α 2 for λ. Proposition 5.2 (b) implies when (λ, α1 ) = 1 then H i (P 1 /B, L λ ) = 0 for all i 0, hence L λ P1 /B is isomorphic to O P1 /B( 1), whose degree is 1 = (λ, α1 ), Proposition 5.2 (c) implies when (λ, α1 ) 0, then H 1 (P 1 /B, L λ ) = 0 and H 0 (P 1 /B, L λ ) has dimension (λ, α1 ) + 1. Hence the line bundle L λ P1 /B is O P1 /B((λ, α1 )). Proposition 5.2 (d) implies when (λ, α1 ) 2, then H 0 (P 1 /B, L λ ) = 0 and H 1 (P 1 /B, L λ ) has dimension (λ, α1 ) 1. Hence the line bundle L λ P1 /B is O P1 /B((λ, α1 )). In fact the proof provides a more general result. Consider the following diagramme with natural maps Z sr+1 P r+1 /B σ r f r+1 Z w[r+1] G/B f r ψ r+1 Z w[r] G/P r+1 π r+1 then we have: Lemma 4.. The relative tangent bundle of ψ r+1 is fr+1(l α ), where L α is the line bundle on G/B associated to the character α, and α is the simple root corresponding to s r+1. 8

9 3 Construction of the Degeneration Consider a sequence w = (s 1, s 2,, s m ) of simple reflections. Let B denote B A and P j denote P j A for 1 j m. Then B w = B A B, A B = B w A and P w = P 1 B P 2 B P m = P w A. Both B w and P w are group schemes over A. Definition 5.. Define the following twisted action of B w on P w over A as follows: [(p 1, p 2,, p m ), x] [(b 1, b 2,, b m ), x] = [(p 1 b 1, φ x (b 1 ) 1 p 2 b 2,, φ x (b m 1 ) 1 p m b m ), x] where φ x : B B is the family of homomorphisms defined in Lemma 2. Lemma 6.. The action of B w on P w over A is free. Prooof: Recall that an action σ : G S X X of a group scheme G over a scheme S on a Scheme X over S is said to be free if the map (σ, π 2 ) : G S X X S X is a closed immersion. [[MFK], Page 10]. We show the map P w A B w P w A P w is injective. Suppose ((p 1, p 2,, p 3 ) (b 1, b 2,, b m ), (p 1, p 2,, p m )) = ((p 1, p 2,, p m) (b 1, b 2,, b m), (p 1, p 2,, p m)) over a point x. Then p 1 = p 1, p 2 = p 2, p m = p m, and (refer Definition 5) p 1 b 1 = p 1 b 1, φ x (b 1 ) 1 p 2 b 2 = φ x (b 1) 1 p 2 b 2, φ x (b 2 ) 1 p 2 b 3 = φ x (b 2) 1 p 2 b 3,, φ x (b m 1 ) 1 p m b m = φ x (b m 1) 1 p m b m. Which successively implies b 1 = b 1, b 2 = b 2, b m = b m. Using the fact that the B orbits in P i are closed embedding, one can show that the map P w A B w P w A P w is a closed immersion. 9

10 Definition 7.. Since the action of B w on P w over A is free, the quotien Z w = P w /B w exists as an algenraic space over A [[KM], Theorem 1.1]. Let π : Z w A denote the defining morphism. Remark 8.. Note that the projection P w[r] P w[r 1] is equivariant for the actions of B w[r] and B w[r 1] on P w[r] and P w[r 1] respectively. Hence this projection descend to give a morphism of the quotient spaces ψ r,a : Z w[r] Z w[r 1],. It is a P r /B fibration. There is also a section σ r,a : Z w[r] Z w[r+1] to the projection ψ r+1,a induced by the inclusion P w[r] = P1 P 2... P r {1} A P 1 P 2... P r+1 A = P w[r+1] Now we describe a scheme structure on the algebraic space Z w. Theorem 9.. The morphism π : Z w A has the following properties: 1. π is a smooth projective morphism 2. The fiber over 1, Zw 1 := π 1 (1) is the BSDH variety Z w and Zw x := π 1 (x) is isomorphic to Z w for x 0 3. Zw 0 := π 1 (0) is a smooth toric variety. Proof of (1) We construct A schemes Y i and smooth morphisms ψ i,a : Y i /B Y i 1 /B, and sections σ i 1,A : Y i 1 /B Y i /B for i = 1, 2,, m inductively. Set Y 0 = B, Y 1 = P 1 and ψ 1,A is the defining morphism from Z w[1] = P 1 /B A. For i 2, define Y i := Y i 1 B P i where the B action is defined by (y, p) b = (yb, φ x (b) 1 p) for y Y i 1, p P i, b B, and x A. There is still an action of B (in fact P i ) on Y i coming from right multiplication on P i and Y i /B = Z w[i]. Note that the P i action is free and Y i /P i = Zw[i 1]. In fact, the scheme Y i is a principal P i bundle over Z w[i 1]. 10

11 We define ψ i,a to be the composite of the following maps ψ i,a : Z w[i] = Y i /B Y i /P i = Yi 1 /B = Z w[i 1] The map sending y to the class [y, 1] from Y i 1 to Y i descend to give σ i 1,A : Y i 1 /B Y i /B Since G is simply connected group, there exists an irreducible rank 2 representation V i of P i which trivially extends over the base A to P i. This gives rise to a rank 2 vector bundle V i on Z w[i 1] such that the B quotient Y i /B = Z w[i] is cannonically isomorphic to P(V i ). We get the following diagramme: Z w[i] = P(V i ) ψ i,a Z w[i 1] Since each ψ i,a is smooth proper morphism as it is isomorphic to the projective bundle of a vector bundle, the composition π = ψ m,a ψ m 1,A ψ 1,A is a smooth projective morphism. Proof of (2): For x 0, consider the map f x : P 1... P m P 1... P m given by f x ((p 1,..., p m )) = (p 1, ξ(x) p 2,..., ξ(x) p m ), where ξ(x) p i is the multiplication in P i. We show that this is a B w equivarient map: f x ((p 1, p 2... p m ) (b 1, b 2... b m )) = f x ((p 1 b 1, φ x (b 1 ) 1 p 2 b 2... φ x (b m 1 ) 1 p m b m )) = (p 1 b 1, ξ(x)φ x (b 1 ) 1 p 2 b 2... ξ(x)φ x (b m 1 ) 1 p m b m ) (Refer Lemma 2 for φ x ) = (p 1 b 1, b 1 1 ξ(x)p 2 b 2..., b 1 m 1ξ(x)p m b m )) = f x ((p 1, p 2... p m )) (b 1, b 2... b m ) This B w equivarient isomorphism descends to give a well defined isomorphism f x : Z w Z x w. Note that the map f 1 is the identity map. Which proves the claim Z 1 w = Z w. Proof of (3): 11

12 Note that Zw[i] 0 can also be viewed as Z 0 w[i] = Y 0 i 1 B (P i /B) where B acts on P i /B via its projection to the maximal torus T. We observe that the action of the maximal torus T on P i /B factors through the action of the multiplicative group on the projective line P i /B via the character α i. We denote this quotient of T by T i. We define T w[r] := T 1 T r, for all 1 r m. Then one see that the action of T T on Z 0 w[i] factors through T w[i]. For i 2, consider the principal B-fibration Y 0 i 1 Y 0 i 1/B = Z 0 w[i 1]. Let E i := Y 0 i 1 B T i ) Z 0 w[i 1] be the principal T i bundle obtained using the associated construction with the quotient homomorphism B T i. Then E i is a T w[i] variety with E i Z 0 w[i 1] is a T w[i 1] equivariant map. Then we have Z 0 w[i] = Y 0 i 1 B P i /B = (Y 0 i 1 B T i ) T i P i /B = E i T i P i /B Now the Theorem follows as Zw[i] 0 has a dense open orbit for the action of T w[i 1] T i = Tw[i]. Hence Zw[i] 0 is a smooth toric variety. Notice that P r+1 /B has two T r+1 -fixed points, one is the B-fixed point called the Schubert point and the other called non-schubert point. These give rise to two sections σ 0 r, σ 1 r : Z 0 w[r] Z 0 w[r+1] The section σ 0 r corresponding to the B-fixed point is called a Schubert section which is the restriction of the section σ r,a to the special fibre. The other T r+1 fixed point of P r+1 /B gives the other section σ 1 r disjoint from the Schubert section. This section will be called non-schubert section. The point σ 0,A (x) (P 1 /B) x = Z x w[1] is called the Schubert point in Z x w[1] The Schubert point of Z x w[r] is defined inductively as the image of the Schubert point under the Schubert section, σ r 1,A Z x w[r 1]. 12

13 A Schubert line in Zw[r] 0 is defined to be any T w[r]-invariant curve containing the Schubert point. More generally we may call a face to be a Schubert face if it contains the Schubert point. The following Lemma is standard. Lemma 10.. Suppose V be a rank 2 vector bundle over a curve C and P(V ) be the projective bundle. Then the sections σ : C P(V ) are in one to one corresspondence with the line bundle quotients Q of V. Moreover the self intersection number of σ(c) in the surface is given by deg Q deg V, where deg denote the degree of the locally free sheaves. We also need the following result which can be extracted from the proof of Theorem 9. We state this as a separate Lemma. Lemma 11.. The section σ i 1,A : Z w[i 1] Z w[i] provides a line bundle quotient V i Q. Let S i be the kernel (a line bundle). For each x A let us denote the restrictions of these bundles on Z x w[i 1] by V x i, S x i respectively. Then Z x w[i] = P(V x i ) and Q x i Z 0 w[i] = P(Q 0 i S 0 i ) with the projection V x i Q x i providing the Schubert section σ 0 i 1(Z x w[i 1] ) for all x A. Moreover the non-schubert section, σi 1(Z 1 w[i 1] 0 ), is provided by the quotient Q 0 i Si 0 Si 0. We can now state a generalization of Lemma 3 to the family. Lemma 12.. Let u = (s 1, s 2 ) be a sequence of simple reflections. Then the self intersection number of the section σ 1,A (Z x s 1 ) in the surface Z x u is (α 2, α 1 ), for all x A. Proof: The above Lemma 10 and Remark 11 shows the the self intersection number of σ 1,A (Z s1 ) does not change in the family. Over a general fiber this 13

14 surface is isomorphic to a Schubert surface. Lemma 3 and Theorem 9(2). For x 0 this follows from 4 Chow group of 1-cycles Given a variety X defined over k we denote the group of 1-cycles on X modulo numerical equivalence by N 1 (X). Let A 1 (X) denote the real vector space N 1 (X) R. Similarly let N 1 (X) be the group of line bundles modulo numerical equivalence and A 1 (X) denote the real vector space N 1 (X) R. Both A 1 (X) and A 1 (X) are finite dimensional by a theorem of Neron-Severi. It is also known that the intersection pairing A 1 (X) R A 1 (X) R is perfect. Let us denote the rational curve Z x w[1] = Zx s 1 by L 1. The fibre of ψ r,x = ψ r,a Z x w[r] over the Schubert point is the Schubert line L r. We index (label) all T w -invariant curves in Zw, 0 which project to a Schubert point in Zw[j] 0 for some j, by m-admissible sequences as follows. For r 2, let I = (i 1,, i j ) be an r 1 admissible sequence and L I be the corresponding labelled curve in Z 0 w[r 1]. Then the T w[r]-invariant curve σ 0 r 1(L I ) in Z 0 w[r] is denoted by the same symbol L I and σ 1 r 1(L I ) is denoted by L Ir, where Ir denote the r-admissible sequence (i 1,, i j, r). The group A 1 (Z x w) is freely generated by the Schubert lines L 1, L 2 L m (cf. [Ba], Lemma 1.1). Hence for any m-admissible sequence I = (i 1, i 2, i r ) we have L I = m j=1 d jl j, for some d j R. In the next proposition we give an explicit formula to write down the coefficients d j. Example 1: In the following pictures T -invariant curves are shown for Z 0 w[1], Z 0 w[2] and Z0 w[3] respectively. 14

15 L L L 1 L 123 L L L 12 L 12 L L 2 7 L Example 2: consider the curve L in Z 0 w[r] for r 9. This line project down to L 3 in Z 0 w[3], i.e ψ 4ψ 5 ψ 6 ψ 7 ψ 8 ψ 9 ψ r (L ) = L 3 and σ 0 r 1 σ 0 9σ 1 8σ 0 7σ 1 6σ 1 5σ 1 4σ 0 3(L 3 ) = L The following observations will be useful in later sections: 1. The schubert section σ 0 j (Z 0 w[j] ) and the non-schubert section σ1 j (Z 0 w[j] ) do not intersect. In the above picture (Example 1), the Schubert surface is placed at the bottom and the non-schubert surface placed at the top. 2. Let I = (i 1, i 2,,, i r ) be an m admissible sequence. Then L I does not exist in Z 0 w[j] for j < i r, Moreover L I lies in the non Schubert section of Z 0 w[j] for j = i r and in the Schubert section of Z 0 w[j] for all j > i r Remark 13.. (i) The indexing set consists of 2 m 1 elements as it corresponds to the set of nonempty subsets of the set {1,, m}. (ii) We show that the total number of T w -invariant curves in Z 0 w is m2 m 1. One can inductively show that the number of T w[r] invariant points, pt r, in Zw[r] 0 is 2r. The T w[r] invariant curves, l r, in Zw[r] 0 can be counted as follows: There are l r 1 T w[r] -invariant curves in the Schubert section σ 0 (Zw[r 1] 0 ) and l r 1 T w[r] -invariant curves in the non-schubert section σ 1 (Zw[r 1] 0 ). Also there are 2 r 1 T w[r] -invariant curves in the fibre which project down to the 2 r 1 15

16 T w[r 1] -invariant points of Z 0 w[r 1]. Thus l r = 2l r 1 + pt r 1 r 2 Now we prove the assertion by induction on m. Clearly the assertion is true when m = 1. Assume the result for m = r, the number T w[r] - invariant curves in Z 0 w[r] is r2 r 1. By the above observation the number T w[r+1] -invariant curves in Z 0 w[r+1] is 2r2r r = 2 r (r + 1). We can also count these T w -invariant curves using our labellings. We have labelled certain curves with non-empty-ordered(increasing order) subsets of the set {1, 2,, m}. Clearly there are 2 m 1 curves with the labelling set starting with 1. Any T w -invariant curves with the labelling set starting with i, i 2 will project down to L i, The curve L i will project down to a T w[i 1] - invariant point (the Schubert point) of Zw[i 1] 0. The number of labelling set starting with i is 2 m i. But Zw[i 1] 0 has 2 i 1 T w[i 1] -invariant points. Any T w -invariant curves projecting down to any of these points other than the Schubert points are not labelled. There are 2 m i T w -invariant curves over each of the T w[i 1] -invariant points, each of these curves is equivalent to one of those labelled curves. The total number of curves (labelled and un-labelled) which are equivalent to one of the labelled curves with labelling set starting with i is 2 m i 2 i 1 = 2 m 1. This is true for all 1 i m. Hence the total number T w -invariant curves is m2 m 1. Lemma 14.. Suppose Y X be smooth projective varieties and C 1, C 2 be two curves in Y. If a 1 C 1 + a 2 C 2 = 0 in A 1 (Y ) then a 1 C 1 + a 2 C 2 = 0 in A 1 (X). In most of our computations on the limiting toric varieties Zw[r] 0 we will work on a suitable surface (refer Lemma 12) Zu 0 Zw[r] 0. The following Lemma is used repeatedly in most of the computation related to curves. Lemma 15.. Let u = (s 1, s 2 ) be a sequence of simple reflections. Let L 1 denote the section σ 0 1(Z 0 s 1 ), L 12 denote the section σ 1 1(Z 0 s 1 ) and L 2 denote the 16

17 fiber (P 2 /B) 0 over the Schubert point in Z 0 u. Then L 12 = L 1 (2, 1)L 2 in N 1 (Z 0 u). Recall that the notation (2, 1) stands for (α 2, α 1 ) Proof: Since the curves L 1, L 2 generate A 1 (Z 0 u) there exist numbers a, b such that L 12 = al 1 + bl 2. Recall that L 1 L 1 = (2, 1) (by Lemma 12), and L 2 L 2 = 0 (because L 2 is a fiber). Clearly L 1 L 12 = 0, L 1 L 2 = 1 = L 2 L 12. By intersecting with L 2, L 12 L 2 = al 1 L 2 + bl 2 L 2, we get 1 = a. Similarly intersecting with L 1, we get 0 = a(2, 1) + b. Thus a = 1 and b = (2, 1) Recall s α (x) := x (x, (α ) )α = x (x, α)α. Definition 16.. Set α j 1 j 2 j r := s α jr s α jr 1 s α j2 (α j 1 ). Proposition 17.. Let w = (s 1,, s m ) be a sequence of simple reflections with m 2. For 2 r m, let I = (i 1, i 2,, i r ) := (i 1, i 2, I ) be an m-admissible sequence and L I be the corresponding curve. Then in the Chow group A 1 (Z 0 w) of 1-cycles, we have: (i) L I = L i1 I (i 2, i 1 )L i2 I (ii) L I = L i1 + r k=2 k d=2 1=j 1 < <j d =k ( 1)d+1 (i jd, i jd 1 ) (i j2, i j1 )L ik (iii) Set d i1 := 1 and d ik = k 1 j=1 d i j (i k, i j ) for 2 k r. Then L I = r j=1 d i j L ij. (iv) d ij = (α i 1 i 2 i j 1, α ij ) for all j > 1 Proof of (i): Note that the curves, L i1 I, L i 2 I and L i 1 i 2 I are edges of a surface that is obtained from the Schubert surface < L i1, L i2, L i1 i 2 > by successively taking sections and hence are isomorphic. Now (i) follows from Lemma 15. Proof of (ii): By (i) L i1 i 2 i r get: = L i1 i 3 i r (i 2, i 1 )L i2 i 3 i r. By induction we L i1 i 3 i r = 17

18 r r k 3 L i1 (i k, i 1 )L ik + ( 1) d+1 (i k, i jd )(i jd, i jd 1 ) (i j1, i 1 )L ik k=3 k=4 d=1 2<j 1 < <j d <k and L i2 i 3 i r = r r k 3 L i2 (i k, i 2 )L ik + ( 1) d+1 (i k, i jd )(i jd, i jd 1 ) (i j1, i 2 )L ik k=3 k=4 d=1 2<j 1 < <j d <k By multiplying the second equation with (i 2, i 1 ) and adding to the first, one can easily check (ii). Proof of (iii): For k = 1, 2, the formula follows directly from definition and (ii). For k 3, the coefficient of L ik in the expansion of L I is given by (ii): k d=2 1=j 1 < <j d =k ( 1) d+1 (i jd, i jd 1 ) (i j2, i j1 ) = k (i jd, i jd 1 ) d=2 = That proves (iii). 1=j 1 < <j d 1 ( 1) d (i jd 1, i jd 2 ) (i j2, i j1 ) k (i jd, i jd 1 )d ijd 1 d=2 Proof of (iv): Proof is by induction on j. Clearly d i2 = (i 2, i 1 ) = (α i2, αi 1 ) = (αi 1, α i2 ). Assume the result for d i1,, d ij 1. Now (αi 1 i 2 i j 1, α ij ) = (s α ij 1 (αi 1 i 2 i j 2 ), α ij ) = (αi 1 i 2 i j 2 (αi 1 i 2 i j 2, α ij 1 )αi j 1, α ij ) = (αi 1 i 2 i j 2, α ij ) (αi 1 i 2 i j 2, α ij 1 )(αi j 1, α ij ) = (αi 1 i 2 i j 2, α ij )+d ij 1 (αi j 1, α ij ) (by inducation hypothesis) = (αi 1 i 2 i j 3, α ij )+d ij 2 (αi j 2, α ij )+d ij 1 (αi j 1, α ij ) = = (αi 1, α i2 )+d i2 (αi 2, α ij )+ d ij 2 (αi j 2, α ij )+d ij 1 (αi j 1, α ij ) = d ij by (iii). 18

19 Corollary 18.. Suppose I = (i 1, i 2,, i r ) with α i1 = α i2. Then r L I = c ij L ij a) c i2 = 1 and for k k > 2, c ik = ( 1) d (i jd, i jd 1 ) (i j2, i j1 ) j=2 d=2 2=j 1 < <j d =k b) c ij = (α i 2 i 3 i j 1, α ij ) for all j > 2 Proof: Replace i 1 by i 2 and use the fact (i 2, i 1 ) = 2 in Proposition 17 (ii) and (iv). Remark 19.. The coefficients c ij in corollary 18 are negative of the coefficients of L I where I = (i 2, i 3,, i r ) Remark 20.. The coefficient of L k in L i1 i 2 i r vanishes for all k {i 1, i 2 i r }. For any m-admissible sequence (i 1, i r, I) the coefficient of L ij and the coefficient of L ij in L i1 i r are the same for all j, 1 j r. in L i1 i ri Remark 21.. For an m-admissible sequence I = (i 1, i 2, i r ), the coefficient d ij of L ij in the expression L I = d ij L ij is the negative of the self intersection of σi 0 j 1(L i1 i j 1 ) in the surface ψ 1 i j (L i1 i j 1 ). From the inductive definition of the coefficients of L I, the coefficients may seem to take arbitrary values but Proposition 17 (iv) shows that these numbers belong to the set {(α, γ ) α S and γ is any dual root. In other words, the coefficients are bounded by coxeter number of the dual root system 5 A basis for A 1 (Z 0 w) The main result of this section is the following basis theorem and the two algorithms to find out this basis from the given sequence w. 19

20 Theorem 22.. Let w = (s 1, s 2,, s m ) be a sequence of simple reflections. Then there exist a set of m linearly independent T w invariant curves, L j (w) 1 j m, of Zw 0 which generate A 1 (Zw) 0 such that every T w invariant curve lies in the Z 0 span of this set. Proof: We choose the generating set inductively. To begin the induction we note that for Zw[1] 0 = P 1, the assertions of the theorem are valid. Suppose we have chosen a generating set L j (w[r]), 1 j r for Zw[r] 0 such that every T w[r] invariant curve in Zw[r] 0 are non-negatively generated by L j(w[r]). Then a generating set for Zw[r+1] 0 can be chosen as L j (w[r + 1]) := { σr(l 0 j (w[r])) if σr(l 0 j (w[r])) 2 0 in ψr+1(l 1 j (w[r])) σr(l 1 j (w[r])) if σr(l 0 j (w[r])) 2 > 0 in ψr+1(l 1 j (w[r])) for 1 j r and L r+1 (w[r + 1]) := L r+1. Here σ 0 r(l j (w[r])) 2 denote the self intersection number in the surface ψ 1 r+1(l j (w[r])). By induction we know that any T w[r] invariant curve in Z 0 w[r] is a positive linear combination of L j (w[r]). First observe from the definition of L j (w[r + 1]), j = 1, r, that the set of curves σ 0 r(l j (w[r])) and σ 1 r(l j (w[r])) are non-negative linear combinations of < L r+1, L r (w[r + 1]), L 1 (w[r + 1]) >. Any T w[r+1] -invariant curve in Z 0 w[r+1] is either equivalent to the fibre L r+1 or lies in either of the sections σ i r(z 0 w[r] ). Now any T w[r+1]-invariant curve in either of the sections σ i r(z 0 w[r] ) are non-negative linear combinations of σi r(l j (w[r])) by induction hypothesis. Hence they lie in the Z 0 span of L j (w[r + 1]). Note that the curve L j (w) is represented by an m-admissible sequence with initial entry j, i.e., L j (w) = L I where I = (j, ). We have seen that any T w invariant curve L I is a linear combination of Schubert lines L i with integer coefficients. But the coefficients can be negative. For example if s 1 = s 2 then L 12 = L 1 2L 2. The idea is to replace L 1 by L 12 in the generating set whenever L 12 = L 1 + d 2 L 2 and d 2 < 0. This 20

21 observation provides an algorithm to obtain the basis. Lemma 23.. Let w = (s 1, s 2,, s m ) be a sequence of simple reflections. 1. If i 2 > 1 is the smallest positive integer such that s i2 = s 1 then L 1 (w[j]) = L 1, j, 1 j i 2 1 and L 1 (w[i 2 ]) = L 1i2. If there is no such i 2 then L 1 (w) = L 1 (w[k]) = L 1, for all k Suppose there exist an i 2 such that s i2 = s 1. If i 3 > i 2 be the smallest positive integer such that c 3 := (i 3, i 2 ) < 0 then L 1 (w[j]) = L 1i2, j, i 2 j i 3 1 and L 1 (w[i 3 ]) = L 1i2 i 3. If there is no such i 3 exists then L 1 (w) = L 1 (w[k]) = L 1i2, for all k i Let L 1 (w[i r 1 ]) = L 1i2 i 3 i r 1 be chosen inductively for r 3. If i r > i r 1 is the smallest positive integer such that c r = (i r, i 2 ) c 3 (i r, i 3 ) c 4 (i r, i 4 ) c r 1 (i r, i r 1 ) < 0 then L 1 (w[j]) = L 1 i 2 i r 1, j, i r 1 j i r 1 and L 1 (w[r]) := L 1i2 i 3 i r. If there is no such i r exists then L 1 (w) = L 1 (w[k]) = L 1i2 i r 1, for all k i r For j > 1, we repeat these procedures for the truncated word [j 1]w. Proof: In other words L j (w) = L 1 ([j 1]w). We obtain L 1 (w) inductively. Clearly L 1 (w[1]) = L 1 gives the required basis for A 1 (Z 0 w[1] ). By Theorem 22 and Lemma 3, L 1(w[2]) is L 1 or L 12 (note that σ 0 1(L 1 ) = L 1 and σ 1 1(L 1 ) = L 12 ) depending on whether (2, 1) is non positive or positive. From the theory of root system we know that (2, 1) is positive only when s 2 = s 1 If s 1 s j, j > 1 in the sequence w = (s 1,, s m ) then L 1 (w[k]) = L 1, k 1 If there exists a j > 1 such that s j = s 1 then let i 2 > 1 be the smallest positive integer such that s 1 = s i2. As above we have L 1 (w[k]) = L 1 k < i 2 and L 1 (w[i 2 ]) = L 1i2. Again by Theorem 22 L 1 (w[i 2 + 1]) is σ 0 i 2 (L 1 (w[i 2 ])) or σ 1 i 2 (L 1 (w[i 2 ])) depending on whether σ 0 i 2 (L 1 (w[i 2 ]) 2 is non positive or positive in the surface ψ 1 i 2 +1 (L 1i 2 ). But σ 0 i 2 (L 1 (w[i 2 ]) 2 is the negative of the 21

22 coefficient of L i2 +1 in L 1i2 (i 2 +1) ( i.e if L 1i2 (i 2 +1) = L 1 + d i2 L i2 + d i2 +1L i2 +1 then σ 0 i 2 (L 1 (w[i 2 ]) 2 is d i2 +1). By Proposition 17(iii) d i2 +1 = 1(1, i 2 + 1) d i1 (i 2, i 2 + 1) = (i 2, i 2 + 1) + 2(i 2, i 2 + 1) = (i 2, i 2 + 1) [as s 1 = s i2 and d i1 = 2] This justifies step 2. A similar argument will work for step 3. Note that c r is the coefficient of L ir in L 1i2 i 3 i r 1 i r, which is negative of the self intersection number, σi 0 r 1(L 1i2 i 3 i r 1 ) 2, in the surface ψ 1 i r (L 1i2 i 3 (i r 1)). Now the proof follows from Theorem 22. The above algorithm can be written using root data related to the algebraic group G. This will also prove the Remark 20. Recall also that the height of a positive root γ = i n iγ i, where γ i s are simple roots, is defined to be the number i n i and is denoted by ht(γ) in a root system. the following lemma. Recall α = x (x, (α ) )α = x (x, α)α. We will use this definion for the dual roots {α } in Definition 24.. Set α j 1 j 2 j r := s α jr s α jr 1 s α j2 (α j 1 ). 2α and the simple reflection s (α,α) α (x) := Let w = (s 1, s 2,, s m ) be a sequence of simple reflections. We give an algorithm to find the subsequence I(w) Let i 2 > 1 be the smallest positive integer such that s i2 = s 1. Let i 3 > i 2 be the smallest positive integer such that ht(α i 2 i 3 ) >ht(α i 2 ). Suppose we have chosen i 1, i r 1 inductively with r 3. Let i r > i r 1 be the smallest positive integer such that ht( α i 2 i 3 i 4 i r ) >ht( α i 2 i 3 i 4 i r 1 ). Theorem 25.. Given a sequence w = (s 1, s 2,, s m ) the Schuber lines L I([j 1]w) = L j (w), 1 j m. Proof: By definition α i 2 i j = s α ij (α i 2 i j 1 ) = α i 2 i j 1 (α i 2 i j 1, α ij )α i j. It is easy to see that (α i 2 i j 1, α ij ) = c ij. α i 2 i 3 i 4 i j 1 ) if and only if c ij Lemma 23. Note that ht(α i 2 i 3 i 4 i j ) >ht( < 0. Now the Lemma follows from the previous 22

23 Remark 26.. Even though the length of the sequence could be very large, the number of steps in the above algorithm can not be very large. For example if the height of the longest root in the dual Weyl group orbit of α1 (i.e the unique dominant weight in the orbit containing α1 ) is n 1 then the number of steps in the algorithm for L 1 (w) will be at most n 1. In other words if L 1 (w) = L 1i1 i r, then r n 1. 6 Extremal Rays and Mori Rays Let X be a normal projective variety defined over k. Then we recall the following definitions (motivated by the definition given for k = C in Page 254 of [W]). Definition 27.. Let NE(X) A 1 (X) be the R 0 cone spanned by effective 1-cycles. A ray R NE(X) is an extremal ray if given Z 1, Z 2 NE(X) such that Z 1 + Z 2 R, then both Z 1, Z 2 R. Definition 28.. If an extremal ray R satisfies R K X < 0, then R is called a Mori extremal ray (also referred as a Mori ray) where K X canonical bundle of X. denote the Lemma 29.. Let X be a variety such that NE(X) is nonnegatively generated by a linearly independent set of effective 1-cycles. Then the rays defined by this generating set are precisely the extremal rays. Proof: Let Z 1,, Z n be the linearly independent set of effective curves generating A 1 (X). First we prove that the rays R 0 Z i s are extremal rays. Suppose a i 0 a iz i + b i 0 b iz i = cz k R 0 Z k. Then a i = b i = 0, for i k as Z i s are linearly independent. Now both a i 0 a iz i = a k Z k and b i 0 b iz i = b k Z k lie in R 0 Z k. Consider any extremal ray generated by R = a i 0 a iz i. If it is not one of the R 0 Z i s, then there are at least two nonzero coefficients, say a k and 23

24 a l. Let R 1 := i l a iz i and R 2 := a l Z l. Then clearly R 1 + R 2 = R lies in R 0 R, but neither R 1 nor R 2 lies in R 0 R, a contradiction. Which proves that R 0 Z i s are the only extremal rays. Theorem 30.. The extremal rays of the toric variety Z 0 w are precisely the curves L j (w). Proof: In view of Lemma 29 it suffices to prove that the effective cone NE(Z 0 w) coincides with the positive convex cone generated by the torus invariant curves L i (w). Since L i (w) form a basis for A 1 (Z 0 w), we can write any effective curve C as a linear combination n i L i (w). For each 1 l l(w), consider the divisor D(a l ) = i l D i (w) + a l Dl (w). Then D(a l ) is ample for all a l > 0 by Theorem 39(a). Hence D(a l ) C = i l n i + a l n l > 0. But i l n i + a l n l > 0 for all a l > 0 implies n l 0. Hence C belongs to the cone generated by L i (w). Now we give a criterion for an extremal ray to be a Mori ray (cf. Page 254 [W]). Recall the following standard lemma: Lemma 31.. Let Z be a complex manifold and X, Y submanifolds of Z intersecting transversally. Let N X/Z denote the normal bundle of X in Z. Then N X/Z X Y = NX Y/Y. If X is a divisor in Z then N X/Z = O Z (X) X. We recall the boundary of a toric variety as the complement of the dense open orbit or equivalently as the union of all torus invariant divisors. Then the boundary of Z 0 w can be inductively shown to be (Zw[r]) 0 := ψr 1 ( (Zw[r 1])) 0 σr 1(Z 0 w[r 1]) 0 σr 1(Z 1 w[r 1]) 0 Then the canonical bundle K Z 0 w[r] = (Z 0 w[r] ) (cf. [O]). Given a Schubert line L r, we compute it s intersection with the boundary components in the following: 24

25 Lemma 32.. For i r 1, (ψ i+1 ) 1 σ 0 i (Z 0 w[i] ) L r, L i+1 = L r. Moreover 1. i < r 1, (ψ i+1 ) 1 σ 0 i (Z 0 w[i] ) L r = 0 = (ψ i+1 ) 1 σ 1 i (Z 0 w[i] ) L r 2. i > r 1, (ψ i+1 ) 1 σ 0 i (Z 0 w[i] ) L r = (i+1, r) and (ψ i+1 ) 1 σ 1 i (Z 0 w[i] ) L r = 0 3. i = r 1, (ψ i+1 ) 1 σ 0 i (Z 0 w[i] ) L r = 1 = (ψ i+1 ) 1 σ 1 i (Z 0 w[i] ) L r Proof: Note that L i+1 (ψ i+1 ) 1 σ 0 i (Z 0 w[i] ), in fact, L i+1 is normal to (ψ i+1 ) 1 σ 0 i (Z 0 w[i] ) and all other Schubert lines are contained in (ψi+1 ) 1 σ 0 i (Z 0 w[i] ) When i r 1, L i+1 L r. Hence L r is in (ψ i+1 ) 1 σi 0 (Zw[i] 0 ). One can easily see that (ψ i+1 ) 1 σi 0 (Zw[i] 0 ) L r, L i+1 = L r. Now by Lemma 31 (ψ i+1 ) 1 σ 0 i (Z 0 w[i] ) L r is the self intersection number L r L r in the Schubert surface L r, L i+1 When i < r 1, the Schubert surface L r, L i+1 maps onto L i+1 with firber L r. Hence L r.l r = 0 in this surface. Which proves 1. When i > r 1, the Schubert surface L r, L i+1 maps onto L r with firber L i+1. Now 2) follows from the Lemma 12. It is clear that L r intersect transversally at the Schubert point and at the non Schubert point of L r transversally. This proves 3. Proposition 33.. K Z 0 w L r = (Zw[r] 0 ) L r = 2 m 1 j=r (j + 1, r) Proof: Note that L r σ 0 j (Z 0 w[j] ), for each j r, we see that σ1 j (Z 0 w[j] ) L r = 0 for j r. Hence we have K Z 0 w L r = ( (Zw)) 0 L r m 1 = (σr 1(Z 0 w[r 1]) 0 + σr 1(Z 1 w[r 1])) 0 L r (ψ j+1 ) 1 (σj 0 (Zw[j])) 0 L r The restriction of the pull back divisor (ψ j+1 ) 1 (σj 0 (Zw[j] 0 )) to L r is isomorphic to the normal bundle of L r in the surface < L r, L j+1 >. Hence the degree, 25 j=r

26 (ψ j+1 ) 1 (σj 0 (Zw[j] 0 )) L r = (j, r) by Lemma 12. Now m 1 K Z 0 w L r = (Zw[r]) 0 L r = 2 (j + 1, r) j=r Lemma 34.. If an extremal ray in Zw[r] 0 is not a Mori ray then the extremal ray lying over this ray cannot be a Mori ray for any Zw[j] 0, j > r. Proof: By induction it suffices to prove for j = r + 1. Let I be an r + 1- admissible sequence such that L I is an extremal ray in Z w[r+1]. Assume ψ r+1 (L I ) is not a Mori ray. Then we have K Z 0 w[r+1] L I = K Z 0 w[r] ψ r+1 (L I ) d where d is the self intersection of the curve L I in the surface ψ 1 r+1(ψ r+1 (L I )). Since ψ r+1 (L I ) is not a Mori ray, it follows that K Z 0 w[r] ψ r+1 (L I ) 0. By the construction of extremal rays (cf. Theorem 22) it follows that d 0. Hence K 0 Z w[r+1] L I 0 as claimed. This leads to a criterion for an extremal ray to be a Mori ray. Theorem 35.. An extremal ray L I is a Mori Ray if and only if there exists an r > 0 such that L I is the Schubert line L r and there is at most one j > r such that (j, r) < 0 and it should be 1. Proof: A Schubert line L r is an extremal ray if and only if (j, r) 0 for all j > r ( refer Lemma 23, Theorem 30). L r is a Mori ray if and only if it is an extremal ray and K Z 0 w L r = j>r (j, r) 2 < 0 i.e., j>r (j, r) 1. So there is at most one j > r such that (j, r) < 0 and it must be 1. Let I = (i 1,, i r ) such that L I = d ij L ij be a non-schubert extremal ray and hence L I = L i1 (w) (by Theorem 30). Notice that (i 2, i 1 ) = 2 and 26

27 (j, i 1 ) 0 for all i 1 < j < i 2 by Lemma 23. Hence K Z 0 L w[i2 i ] 1 i 2 = (j, i 1 ) 2 + (i 2, i 1 ) = i 1 <j<i 2 i 1 <j<i 2 (j, i 2 ) 0 This shows L i1,i 2 Mori ray in Zw. 0 is not a Mori ray in Z 0 w[i 2 ]. Now by Lemma 34 L I is not a Recall that a smooth projective variety is called Fano if it s anti-canonical bundle is ample. Corollary 36.. For a ginve w = (s 1, s 2,, s m ) The toric variety Zw 0 is Fano if and only if all Schubert lines are Mori. Prooof: Suppose all the L i s are Mori, then by definition of Mori, they are all extremal and hence by Theorem 30 and Theorem 22 they generate all the torus invariant lines positively. Now the ampleness of K Z 0 w follows from Toric Nakai Criterion. Conversely if Zw 0 is Fano then L j (w) (refer 22) are Mori. By Theorem 35 L j (w) is the Schubert line L j. Thus all Schubert lines are Mori. Example 3: G = SL n+1 and the sequence w 0 = (1, 2,, n, 1, 2, n 1,.1, 2, 1) is a reduced expression for the longest element w 0 of the Weyl Group. Then there are exactly n Mori rays which are: L n, L n+n 1,, L n+n 1+n 2+ n r,, L n(n+1)/2. 7 Ample Cone Let w = (s 1, s 2,, s m ) be a sequence of simple reflections and Z w[i] be the intermediate BSDH variety. Let f i : Z w[i] G/B be the B-equivariant map. Let L(ω j ) denote the line bundle on G/B corresponding to the fundamental weight ω j. 27

28 Let ψ i := ψ i+1 ψ m : Z w Z w[i] be the composite projection. Then define L j := (ψ j ) (fj (L(ω j )). Then Lauritzen and Thomsen [LT] have proved that L j for j = 1,, m form a basis for the Picard group of line bundles, hence a basis for A 1 (Z w ). In fact they also proved that the ample cone is the strict positive cone generated by L j. Lemma 37.. L j L r = { 0 if j < r or α j α r 1 if j r and α j = α r Proof: For j < r, choose a section of L j that does not contain the Schubert point of Z w[j]. Then the inverse image of this section under ψ j does not intersect with the Schubert line L r and hence the L j L r = 0. For j r, f m restricts to l r as an embedding wth image is P r /B. Moreover, L j restricted to L r can be identified with the restriction of line bundle L(ω j ) to P r /B. Now the theorem follows from the fact that L(ω j ) has degree 0 if α r α j it is 1 if α r = α j The boundary components (ψa) j 1 σ j 1,A (Z w[j 1] ), 1 j m, are simple normal crossing A divisors in Z w. For each x A, these divisors form a basis for the Picard group P ic(zw). x In particular L j can be expressed uniquely as linear combination of these boundary divisors of Zw 1 = Z w. L j = j a ij (ψ i ) 1 σ i 1 (Z w[i 1] ) i=1 Now consider the following relative divisor L j,a := j a ij (ψa) i 1 σ i 1,A (Z w[i 1] ) i=1 We denote the line bundle L 0 j given by the restriction of this relative 28

29 divisor to the special fiber at 0. Now by using Lemma 32, we can show an analogue of Lemma 37 in the toric variety for the line bundle L 0 j. Lemma 38.. A relative line bundle L given by m i=1 a i(ψ i A) 1 σ i 1,A (Z w[i 1] ) on Z w represents a relatively ample bundle if and only if L 0 L j (w) > 0 for all j, where L 0 is the line bundle given by m i=1 a i(ψ i ) 1 σ i 1 (Z 0 w[i 1] ) Proof: Ampleness being an open condition in a flat family to check the relative ampleness it sufices to check ove the special fiber Z 0 w. Note that by our choice of L j (w) any torus invariant curve can expressed as non negative linear combination of these L j (w) (refer 22). Now the stated condition is equivalent to the Toric Nakai Criterion for ampleness for a smooth toric variety. Theorem 39.. The ample cone of Z 0 w is a naturally a subcone of the ample cone of the BSDH variety Z w Proof: Using the basis (ψa) i 1 σ i 1,A (Z w[i 1] ) for the relative P ic(z w ) and the openness of the ampleness one can identify the ample cone of Zw 0 as a subcone of the ample cone of Zw. 1 We would like to give more computable comparison of this sub cone in terms of the natural generators of the ample cone of the BSDH variety. For this purpose let us denote the coeffecients of L j (w) by d ij ([j 1]w) for j = 1, m. Then we have L k (w) = L k + d ij ([k 1]w)L ij i j i j >k Note that d ij ([k 1]w) < 0 for all i j. Now consider an ample line bundle L = a i L i. Then L L k (w) = a k + a i d ij ([k 1]w) 29

30 where the sum is taken all indices i, i j > k and α i = α ij. We can prove that a m > 0 as L L m (w) = a m. By (downward) induction we assume that a i > 0 fo all i > k. then we observe that L L k (w) = a k + a i d ij ([k 1]w) > 0 implies a k > a i d ij ([k 1]w) > 0 This infact proves that the line bundle L is ample on the BSDH variety by the Theorem of Lauritzen and Thomsen. This gives an inclusion of the ample cone. Acknowledgement: The authors would like to thank the referee for many valuable comments and suggestions. Appendix 1. L i1 i 2 i 3 i 4 i 5 = L i1 i 3 i 4 i 5 (i 2, i 1 )L i2 i 3 i 4 i 5 = L i1 i 4 i 5 (i 3, i 1 )L i3 i 4 i 5 (i 2, i 1 )L i2 i 4 i 5 + (i 3, i 2 )(i 2, i 1 )L i3 i 4 i 5 = L i1 i 4 i 5 (i 2, i 1 )L i2 i 4 i 5 + [ (i 3, i 1 ) + (i 3, i 2 )(i 2, i 1 )]L i3 i 4 i 5 = L i1 i 5 (i 4, i 1 )L i4 i 5 (i 2, i 1 )[L i2 i 5 (i 4, i 2 )L i4 i 5 ] [ (i 3, i 1 ) + (i 3, i 2 )(i 2, i 1 )][L i3 i 5 (i 4, i 3 )L i4 i 5 ] = L i1 i 5 (i 2, i 1 )L i2 i 5 + [ (i 3, i 1 )+(i 3, i 2 )(i 2, i 1 )]L i3 i 5 + [-(i 4, i 1 )+(i 4, i 2 )(i 2, i 1 )+ (i 4, i 3 )(i 3, i 1 ) (i 4, i 3 )(i 3, i 2 )(i 2, i 1 )]L i4 i 5 = L i1 (i 5, i 1 )L i5 (i 2, i 1 )[L i2 (i 5, i 2 )L i5 ] + [- (i 3, i 1 ) + (i 3, i 2 )(i 2, i 1 )][L i3 (i 5, i 3 )L i5 ]+ [ (i 4, i 1 ) + (i 4, i 2 )(i 2, i 1 ) + (i 4, i 3 )(i 3, i 1 ) (i 4, i 3 )(i 3, i 2 )(i 2, i 1 )][L i4 (i 5, i 4 )L i5 ] = L i1 (i 2, i 1 )L i2 +[- (i 3, i 1 ) + (i 3, i 2 )(i 2, i 1 )]L i3 + [ (i 4, i 1 ) + (i 4, i 2 )(i 2, i 1 ) + (i 4, i 3 )(i 3, i 1 ) (i 4, i 3 )(i 3, i 2 )(i 2, i 1 )]L i4 +[ -(i 5, i 1 )+(i 5, i 2 )(i 2, i 1 )+(i 5, i 3 )(i 3, i 1 )+ ((i 5, i 4 )(i 4, i 1 ) (i 5, i 3 )(i 3, i 2 )(i 2, i 1 ) (i 5, i 4 )(i 4, i 2 )(i 2, i 1 ) (i 5, i 4 )(i 4, i 3 )(i 3, i 1 )+ (i 5, i 4 )(i 4, i 3 )(i 3, i 2 )(i 2, i 1 )]L i5 Let L i1 i 2 i 3 i 4 i 5 := d i1 L i1 + d i2 L i2 + d i3 L i3 + d i4 L i4 + d i5 L i5 30

31 2. Suppose α i1 = α i2. Then we replace i 1 with i 2 and use (i 2, i 1 ) = 2 in the above expression to get L i1 i 2 i 3 i 4 i 5 = L i2 2L i2 + [ (i 3, i 2 ) + 2(i 3, i 2 )]L i3 + [ (i 4, i 2 ) + 2(i 4, i 2 ) + (i 4, i 3 )(i 3, i 2 ) 2(i 4, i 3 )(i 3, i 2 )]L i4 +[ -(i 5, i 2 )+2(i 5, i 2 )+(i 5, i 3 )(i 3, i 2 )+(i 5, i 4 )(i 4, i 2 ) 2(i 5, i 3 )(i 3, i 2 ) 2(i 5, i 4 )(i 4, i 2 ) (i 5, i 4 )(i 4, i 3 )(i 3, i 2 ) + 2(i 5, i 4 )(i 4, i 3 )(i 3, i 2 )L i5 = L i2 +(i 3, i 2 )L i3 + [(i 4, i 2 ) (i 4, i 3 )(i 3, i 2 )]L i4 + [(i 5, i 2 ) (i 5, i 3 )(i 3, i 2 ) (i 5, i 4 )(i 4, i 2 ) + (i 5, i 4 )(i 4, i 3 )(i 3, i 2 )]L i5 Let L i1 i 2 i 3 i 4 i 5 := c i2 L i2 + c i3 L i3 + c i4 L i4 + c i5 L i5 3. α i 1 : (α i 1, α i2 ) = (α i2, α i 1 ) = (i 2, i 1 ) = d i2 α i 1 i 2 = s αi2 (α i 1 ) = α i 1 (α i 1, α i2 )α i 2 : (α i 1 i 2, α i3 ) = (i 3, i 1 ) (i 3, i 2 )(i 2, i 1 ) = d i3 α i 1 i 2 i 3 = s α i3 (α i 1 i 2 ) = α i 1 (α i 1, α i2 )α i 2 (α i 1 (α i 1, α i2 )α i 2, α i3 )α i 3 = α i 1 (α i 1, α i2 )α i 2 (α i 1, α i3 )α i 3 + (α i 1, α i2 )(α i 2, α i3 )α i 3 : (α i 1 i 2 i 3, α i4 ) = (i 4, i 1 ) (i 4, i 2 )(i 2, i 1 ) (i 4, i 3 )(i 3, i 1 )+(i 4, i 3 )(i 3, i 2 )(i 2, i 1 ) = d i4 It is easy to check that (α i 1 i 2 i 3 i 4, α 5 ) = d i5 and (α i 2, α 3 ) = c i3, (α i 2 i 3, α 4 ) = c i4, (α i 2 i 3 i 4, α 5 ) = c i5 31

32 References [Ba] [C] [D] [GL] [GK] Charles M. Barton, Tensor Product of Ample Vector Bundles in Characteristic p Amer. J. Math. 93 (1971) P. Caldero, Toric Degenerations of Schubert varieties, Transformation Groups, Vol. 7, No. 1, (2002), M. Demazure, Desingularization des varietes de Schubert generalisees, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1974) N. Gonciulea and V. Lakshmibai, Degenerations of flag and Schubert varieties to toric varieties, Transformation Groups Vol. 1, No 3, (1996). M. Grossberg and Y. Karshon, Bott towers, complete integrability, and the extended character of representations, Duke Math. J. 76 (1994), no. 1, [H] R. Hartshorne, Algebraic Geometry Springer, 1977, [J] J. C. Jantzen, Representations of Algebraic groups, Pure and Applied Mathematics, 9, Academic Press, [KM] S. Keel, S. Mori, Quotients by Groupoid, Annals of Mathematics, 145, (1997) [K] [LT] G. R. Kempf, Linear Systems of Homogeneous Spaces, The Annals of Mathematics, Second Series, Vol 103, No. 3, (1976) N. Lauritzen, J. F. Thomsen, Line bundles on Bott-Samelson varieties, J. Algebraic Geom. 13 (2004), no. 3, [MFK] D. Mumford, J. Fogarty, F. Kirwan, Goemetric Invariant Theory, Springer Verlag [O] T. Oda, Convex Bodies and Algebraic Geometry, Ergeb. Math, (15) Springer Verlag,