Directory UMM :Journals:Journal_of_mathematics:OTHER:

Rational Curves on

Homogeneous Cones

Nicolas Perrin

Received: October 18, 2004 Revised: December 20, 2004

Communicated by Thomas Peternell

Abstract. A homogeneous coneX is the cone over a homogeneous variety G/P embedded thanks to an ample line bundle L. In this article, we describe the irreducible components of the scheme of mor-phisms of classα∈A1(X) from a rational curve to X.

The situation depends on the line bundle L : if the projectivised tan-gent space to the vertex contains lines then the irreducible components are described by the difference between Cartier and Weil divisors. On the contrary if there is no line in the projectivised tangent space to the vertex then there are new irreducible components corresponding to the multiplicity of the curve through the vertex.

2000 Mathematics Subject Classification: 14C05, 14M17.

Keywords and Phrases: homogeneous cone, scheme of morphisms, rational curves.

In this text we study the scheme of morphisms from P1 _{to any homogeneous} cone that is to say a coneX over a homogeneous varietyG/P.

This study is motivated by the more general problem of describing the irre-ducible components of the scheme of morphisms from P1 _{to a variety X}

that the situation is more complicated than in the minuscule case but still has a nice description in terms of the combinatorial data ofG.

More precisely, let G/P be a homogeneous variety and let L be a very ample divisor on G/P. We may embed G/P in P_(H0_{L). If V is a n-dimensional} vector space, it defines a linear subspace P_{(V) in}P_(H0_{L⊕V). Let us denote} by X = CL,n(G/P) the cone above G/P whose vertex is P(V). Now let U

be the open subset of X complementary toY = P_{(V). We have a surjective}

morphism (see paragraph 1):

s: Pic(U)∨_→A1(X).

For any classα∈A1(X), we can consider the following morphism: i: a

s(β)=α

Hom_β(P1_{, U)→}Hom_α(P1_{, X)}

where Hom_α(P1_{, X) is the scheme of morhisms f :}P1 _{→ X withf∗[}P1_{] =α}

and Hom_β(P1_{, U) is the scheme of morhisms g :} P1 _{→ U such that [g] = β}

where [g] is the linear functionL7→deg(g∗_{L) on Pic(U). AsY =X\U lies in}

codimension 2, we expect the image of this morphism to be dense. For example we prove in [P2] that it is true for X a minuscule Schubert variety and U the smooth locus.

In our case the situation will be more complicated. Let us first describe the “expected” components in the case wherei is dominant. In this case we may apply the results of [P1] to prove thatHom_β(P1_{, U) is irreducible as soon as}

it is non empty and the images of the irreducible varieties Hom_β(P1_{, U) will} give the irreducible components ofHom_α(P1_{, X). The expected components} are thus indexed by the subsetne(α) of Pic(U)∨ _{given by elementsβ such that}

s(β) =αandHom_β(P1_{, U) is non empty.}

This set can be discribed in terms of roots: the ample divisorLis a dominant weight in the facet of the parabolic P. An element α∈ A1(X) is completely determined byα·L=d∈Z_{. Denote by}neB(α) the set of all elementsβ in the

cone generated by the positive roots such that hβ∨_{, Li=d. This is a subset}

ofA1(G/B). Thenne(α) is its image inA1(G/P) (see paragraph 1 for a more details). We prove the

THEOREM0.1. — Let R be the root lattice.

(ı) If L(R) = Z_{, then the irreducible components of the scheme of morphisms}

Hom_α(P1_{, X)are indexed byne(α).}

(ıı)IfL(R)6=Z_{(i.e. if we haveL > c1(TG/P)), then the irreducible components} of the scheme Hom_α(P1_{,C(G/P))are indexed by} a

α′≤α

ne(α′_).

We will see (paragraph 1) that A1(X) ≃ Z _{so that α}′ _{≤α in A1(X) means}

meet the vertex of the cone. The integerα−α′ _{is then the multiplicity of the}

curve at the vertex.

Remark 0.2. — (ı)The condition L(R) =Z_{is exactly equivalent to the fact} that there exists lines on G/P embedded with L. In other words there exists lines in the projectivized tangent cone to the singularity.

We studied in [P2] the same problem for minuscule Schubert varieties where the multiplicity in the singularity did not appear. If one consider more gen-eraly quasi-minuscule Schubert varieties of non minuscule type (see [LMS] for a definition, the case of quasi-minuscule Schubert varieties of minuscule type should be very similar to the case of minuscule Scubert varieties) we recover this condition of the existence of lines in the projectivised tangent cone to the singularity.

(ıı)IfP =B is a Borel subgroup and if we choose forLthe Pl¨ucker embedding (or equivalentlyL=ρas a weight whereρis half the sum of the positive roots) then L(R) = Z _{and the set ne(α) is in bijection with the set of irreducible} integrable representations of level exactly α·L of the affine Lie algebraˆg(see paragraph 1 for the general case).

Here is an outline of the paper. In the first paragraph we define the surjective mapsof the introduction and the setne(α) for a homogeneous coneX. In the second paragraph we study the scheme of morphisms fromP1_{to the blowing-up}

e

X of the coneX and prove a smoothing result. In the last paragraph we prove our main result.

The key point as indicated above is to study the surjectivity of the map i that is to say to study the following problem: can any morphismf :P1_→X

be factorised in U (modulo deformation). We do this by lifting f in feon Xe and the problem becomes: does the lifted curve feof a general curve f meet the exceptional divisorE. If it is the case then we add a “line” Γ⊂E(this is possible only whenL(R) =Z_{) with Γ·E=−1 and smooth the unionf}e₍P1_)∪Γ. The intersection with E is lowered by one in the operation. We conclude by induction on the number of intersection offewithE.

We end with a discussion on the dimensions of the components, in particular the variety Hom_α(P1_{, X) is equidimensional if and only ifL =} 1

2c1(G/P) or L=c1(G/P).

1 Preliminary

In this paragraph we explain the results on cycles used in the introduction. We describe the surjective morphism s: Pic(U)∨ _{→A1(X) and define the set of}

IfXis a scheme of dimensionn, we denote byZ∗(X) the group of 1-cycles onX

and byZ≡

∗(X) andZ∗r(X) the subgroups of cycles trivial for the numerical and

rational equivalence. Let us denote by N∗(X) andA∗(X) the corresponding

quotients. The Picard group is the image inAn−1(X) of the subgroup of Cartier

divisors inZn−1(X).

LEMMA 1.1. — Let X =CL,n(G/P) be a cone over a homogeneous variety

G/P then

(ı) Pic(X)≃N1_(X), (ıı)A1(X)≃N1(X).

In particular we have A1(X)≃Pic(X)∨ _{and they are isomorphic toZ.}

Proof. Consider the decompositionV ⊕H0_{L. The following group} G′ ₌

µ

GL(V) Hom(H0_{L, V)}

0 G

¶

acts onX and the unipotent part U(G′_{) acts onX with finitely many orbits.}

Remark that it is not the case if we only take the unipotent partUofGL(V)×G. Indeed, take for example V =C _{andG=GL(V) =GL1(}C_{). ThenG/P is a}

point and X is a projective line but U is the trivial group and has infinitely many orbits inX. The same problem appears in the general case.

(ı) Thanks to the results of [FMcPSS] the groupsA∗(X) are free generated by

invariant subvarieties. The Picard group is contained in An−1(X) and is in

particular free. Thanks to [Fu] Example 19.3.3. this implies that Pic(X) ≃

N1_(X).

(ıı) The results of [FMcPSS] also imply that A1(X) is generated by the one-dimensional invariant subvarieties. The only such subvariety is

• the fibre of the cone over the 0-dimensional orbit inG/P if dimV = 1;

• the 1 dimensional orbit inP_{(V) if dimV ≥2.}

We get the isomorphism A1(X) ≃ Z _{with this variety as generator. This}

generator is clearly numerically free (for example its degree is 1) so we get the result.

The duality comes from general duality betweenN1(X) andN1_{(X). ¤}

LetU be the smooth locus ofX, it is also the dense orbit underG′ _{inX. Let}

Y be the complementary of U in X, it is of codimension at least 2 (at least when dim(G/P) >0). This in particular implies that Pic(U) = An−1(U) ≃

An−1(X). We now have the following inclusion:

Pic(X)⊂An−1(X)≃Pic(U)

giving the surjection

DEFINITION1.2. — Let α∈A1(X). We define the setne(α)⊂An−1(X)∨. Let us make the identification An−1(X)≃Pic(U). The elements of ne(α)are the elementsβ ∈Pic(U)∨ _{such thats(β) =αand there exists a complete curve}

C⊂U with [C] =β as a linear form on Pic(U) (β is effective).

Let us describe ne(α) explicitly: the smooth part U is a vector bundle over G/P. In particular we have Pic(U)≃Pic(G/P).

Let us fixT a maximal Torus inG, fixB a Borel subgroup containingT and suppose that B ⊂P. Let us denote by ∆ the set of all roots, by ∆+ _(resp. ∆−_{) the set of positive (resp. negative) roots and byS the set of simple roots}

associated to the data (G, T, B).

Denote byg,tandpthe Lie algebras ofG,T andP and define

α(p) =nα∈S /gα⊂p andg−α6⊂p

o .

Now set t(p)∗ _{as the subvector space oft}∗ _{generated by the roots inα(p), we}

have

Pic(G/P)≃t(p)∩Q

where t(p) is the dual of t(p)∗ _{in t and Q is the weight lattice. The Picard}

group ofX in Pic(U)≃Pic(G/P)≃t(p)∩Qis given by the the line generated byλ(the weight associated toL). We have

Pic(U)∨_≃t∗_/t(p)∗_∩R

where R is the root lattice. Furthermore, an element β ∈ Pic(U)∨ _{gives an}

effective element if and only if it is in the image of the cone generated by positive roots i.e. in the conet∗_/t(p)∗_∩R+ _{(see [P1]). Then we have}

ne(α) =©β ∈t∗/t(p)∗_∩R+ _{/ hβ}∨_{, λi=α·L}ª

where the integer hβ∨_{, λiis well defined becauseλ∈t(p)∩Q.}

Example 1.3. — Choose for L (or for λ) the smallest ample sheave on X. This is possible: the picard group Pic(U) =t(p)∩Qis a direct sum of weight lattices of semi-simple Lie algebras (gi)i∈[1,r]. We just have to take

λ= X

i∈[1,r] ρi

whereρi is half the sum of positive roots ingi.

Let us denote byirˆgi(ℓ)the set of isomorphism classes of irreductible integrable

representations of level exactelyℓ of the affine Lie algebra ˆgi. Then we have

ne(α) = Y

ℓ₁+···+ℓr=α·L

In particular if P =B is a Borel subgroup of Gthen r= 1 and G1=G and we recover the example of the introduction:

ne(α) =irˆg(α·L).

Remark 1.4. — (ı)The scheme Hom_α(P1_{, X) is the scheme of morphisms}

from P1 _{toX of classα(for more details see [Gr] and [Mo]).}

In general, this will just mean that α ∈ A1(X) and that f∗[P1] = α but sometimes (in particular in the introduction for the open part U) we consider

α∈Pic(X)∨ _{and the class of a morphismf :P}1 _{→X will be the linear form} Pic(X)→ Z _{given by L 7→ deg(f}∗_{L). In the case of a homogeneous cone X} the two notion coincide because of the previous lemma.

(ıı) If X is a variety, α∈A1(X)and F a vector bundle on X we will denote

α·F= Z

α

c1(F)by abuse of notation.

2 Resolution

Recall that we denote byX the coneCL,n(G/P). LetXe be the blowing-up of

X inP_{(V). It is smooth and isomorphic to}

P_{G/P((V ⊗ OG/P)⊕L).}

Let us denote by p the projection from Xe to G/P and by π : Xe → X the blowing-up. The morphism phas natural sections given by points ofP_{(V) or}

equivalently by surjective morphismsL⊕(V ⊗ OG/P)→V ⊗ OG/P → OG/P.

2.1 Cycles on Xe

LEMMA 2.1. — (ı) Rational and numerical equivalences coincide on Xe. In particular we haveA1(X)e ≃Pic(X)e ∨_≃A

n−1(Xe)∨.

(ıı)We havePic(Xe)≃Pic(G/P)⊕Z_{with the factor}Z_{generated by ap-relative} ample class.

Proof. (ı) Rational and numerical equivalence coincide on G/P. Moreover the fibration in projetive spaces Xe → G/P has sections so that rational and numerical equivalences coincide onX. This in particular implies that Pic(e Xe) = An−1(X) =e N1(Xe) andA1(Xe) =N1(Xe) and the duality follows.

(ıı) The varietyXe is aPn_{-bundle overG/P with sections so we get that}

with the factor Z _{generated by a p-relative ample class, the relative tangent}

sheafTp ofpisn+ 1 times this class on the factorZ. ¤

Any element αe∈A1(Xe)≃Pic(Xe)∨ _{is given by the class β =p}

∗αe ∈A1(G/P)

and the relative degreed=αe·Tp. We will use the notationℓ=β·L=αe·p∗L.

Let us denote by E the exceptional divisor on X, it is a triviale Pn−1 _bundle overG/P given by the surjectionL⊕(V⊗ OG/P)→V⊗ OG/P. Then we have:

e

α·E= d−nℓ n+ 1 , it has to be an integer so thatd≡nℓ modn+ 1.

Let us consider the following morphism still denoted byp: p:Hom

e

α(P1,X)e →Homβ(P1, G/P).

PROPOSITION 2.2. — Thanks to the morphism p, the schemeHom

e α(P1,Xe) is an open subset of a projective bundle overHom_β(P1_{, G/P).}

Proof. This generalises proposition 4 of [P1] in the case where the relative degreeαe·Tpis negative. This is possible because the vector bundle associated

to thePn _{fibration has a decompositionL⊕(V⊗ OG/P). We only describe the}

fibers, for the structure of projective bundle see [P1] proposition 4.

Let f : P1 _{→ G/P, we have to calculate the fiber of pabove f. The fiber is}

given by sections of thePn_-bundlef∗_{(p) :P}

P1((V⊗ O_P1)⊕ O_P1(ℓ))→P1whose relative degree isd. In other words the fiber is given by surjections (V⊗ O_P1)⊕ O_P1(ℓ)→ O_P1(x) modulo scalar multiplication whered= (n+1)x−ℓ. The fiber is therefore isomorphic to an open subset ofP_(Hom((V⊗OP1)⊕O_P1(ℓ),O_P1(x)). Let us remark that if Hom_β(P1_{, G/P) is not empty then we have ℓ≥0 and}

in this caseHom

e

α(P1,Xe) is not empty if and only ifx≥0 whenn≥2 and if

and only ifx= 0 or x≥ℓwhen n= 1. In terms ofdthis means thatd=−ℓ or d≥nℓ ifn= 1 and d≥ −ℓ ifn≥2. In any cases, if Hom

e

α(P1,X) is note

empty thenx≥0. There are two cases:

• Ifx < ℓ then any section is included in the exceptional divisor and the dimension of the fiber is:

n

n+ 1(ℓ+d) +n−1.

• Ifx≥ℓthen the fiber is of dimensiond+n. ¤

Let αe ∈ A1(X) such thate Hom

e

α(P1,X) is not empty. This is equivalent toe

the fact that β ∈ A1(G/P) is positive (see [P1], it is equivalent to the fact that Hom_β(P1_{, G/P) is non empty) and such thatd=−ℓor d≥nℓifn= 1,}

COROLLARY 2.3. — The scheme Hom

e

α(P1,Xe)is irreducible of dimension •

Z

e α

c1(T_Xe) + dim(X)e if d≥nℓ

•

Z

e α

c1(T_Xe) + dim(X)e −αe·E−1 ifd < nℓ.

Proof. We just use the preceding proposition and the fact proved in [P1] that the scheme Hom_β(P1_{, G/P) is irreductible of dimension} R

βc1(TG/P) +

dim(G/P). Remark that in the last case we have nℓ > d so that the dimension of Hom

e

α(P1,Xe) is still greater than the expected dimension

Z

e α

c1(T_Xe) + dim(Xe). ¤

2.2 Smoothing curves onXe

In this paragraph we will prove some results on curves onXe. PROPOSITION 2.4. — Assume thatL(R) =Z_.

Letαe∈A1(X)e ,fe∈Hom

e

α(P1,X)e such thatfe(P1)6⊂Eandαe·E >0. Assume that the image of p◦fe:P1_{→G/P is not a line in the embedding given by L.} Then there exists a deformation fe′ _{of feand a curve Γ ⊂Xe contracted by π} withΓ·E=−1 such that the curvefe′_(P1_{)∪Γcan be smoothed. The smoothed}

curve is the image of a morphismfb:P1_→Xe_.

Proof. Let (x, v)∈E≃G/P×P_{(V) be a point in the intersectionf}e₍P1_)∩E.

Let us first remark that we may assume (after deformation) that fe(P1_{) is a}

nodal curve. Indeed, because p◦fe: P1 _{→ G/P is not a line, this implies in}

particular that G/P is not P1 _{so it is of dimension at least 2. The results of}

[P1] prove that a general curve inG/P is nodal and so isfe(P1_{) iff}e_{is general.}

LEMMA 2.5. — There exists a deformationfe′ offeand a rational curve Γin

e

X such that [Γ]·E=−1,[Γ]·L= 1and meeting fe′_(P1_{) in exactly one point.} Proof. Let us consider the lines in G/P that is to say the rational curves Γ′

inG/Psuch that [Γ′_{]·L= 1. Such curves exist because we haveL(R) =Z. Let}

Γ′ _{be such a line passing throughp(x, v) =x∈G/P and let Γ be the section}

of Γ′ _{in E given by the pointv ∈P(V). This curve is contracted byπ to the}

pointv∈P_{(V), its intersection withE is given by−[Γ}′_]·L=−1.

As we assumed thatp◦f(eP1_{) is not a line then Γ}′ _{meets p◦fe(P}1_{) in a finite} number of points: x and other points (xi). The morphism feis given by a

section of the projective bundle overp◦fethat is to say by a surjection

s: (V ⊗ OP1)⊕ OP1(ℓ)→ OP1 µ

d+ℓ n+ 1

To deformfewe can deform this surjectionsins′_{such that atx, we haves}′ valid for any smooth projective variety:

THEOREM2.7. — LetZ be a smooth projective variety and letC be a nodal curve in Z. Assume that the cohomology groupH1_T

Z|C is trivial then C can be smoothed.

Asfe(P1_{) is nodal and thanks to the previous lemma we know thatf}e′_(P1_)∪Γ is nodal. We just have to prove that the cohomology groupH1_(T

e

X|fe′(P1)∪Γ) is trivial. We have the exact sequences

0→Tp→TXe →p ∗_T

G/P →0 and 0→ Ofe′_(P1₎(−Q)→ Ofe′_(P1₎∪Γ→ OΓ→0 where Qis the intersection point of fe′_(P1_{) and Γ. We just have to prove the} vanishing of the following cohomology groups:

H1(p∗_T

G/P|Γ) ; H1(p∗TG/P|fe′(P1)(−Q)) ; H

1_{(Tp|Γ) and H}1_(Tp|

e

f′(P1)(−Q)). The first two groups are respectively equal to H1_(T

G/P|Γ′) and

H1_(T

G/P|_p(fe′_(P1))(−Q)) where we denoted Γ

′ _{= p(Γ). They are trivial}

because TG/P is globally generated and Γ′ andp(fe′(P1)) are rational curves.

Let us denote byOp(1) the tautological quotient of the projective bundle asso-ciated to (V⊗OX)⊕L, the relative tangent sheaf is given byTp= Coker(OXe →

This proves that the group H1_{(Tπ|Γ) vanishes. Furthermore, since feexists} we must have _nd+₊₁ℓ ≥ 0 (see proposition 2.2) and d−nℓ

n+1 = αe·E > 0 so that H1_(Tp|

e

3 Homogeneous cones

Recall that we denote by X the coneCL,n(G/P). In this paragraph we study

the irreducible components of the scheme Hom_α(P1_{, X) where α ∈ A1(X).}

Recall thatA1(X)≃Z_{and under this identificationαis just the degree of the}

corresponding curve.

3.1 The case L(R) =Z

THEOREM3.1. — Assume that L(R) = Z_{, let α ∈ A1(X) and f ∈}

Hom_β(P1_{, X). Then there exists a deformation f}′ _{off such thatf}′ _{does not} meet the vertex P_{(V)of the cone X.}

Proof. Let us begin with the following:

LEMMA 3.2. — Let f ∈Hom_α(P1_{, X)such thatf factors through the vertex}

P_{(V) of the cone. Then there exists a deformation f}′ _{of f in Hom}

α(P1, X) such that f′_(P1_{)does not factor through the vertex.}

Proof. Let x∈ G/P and consider the linear subspace generated by xand

P_{(V). It is a projective space contained inX containing}P_{(V) as a hyperplane}

and containingf(P1_{). In this projective space we can deform the morphism f}

so that it does not factor throughP_{(V) any more. ¤}

A general morphism f ∈ Hom_α(P1_{, X) does not factor through the vertex}

P_{(V) of the cone so it can be lifted in a morphismf}e_:P1_{→Xe. Letαe∈A1(Xe)} the class offe, we haveπ∗αe=α. Becausef does not factor through the vertex,

the morphismfedoes not factor through the exceptional divisorE so we have: e

α·E ≥ 0. If αe·E = 0, thenfe(P1_{) does not meet E thus f does not meet}

the vertex and we are done. Let us assume that α_e·E > 0. We proceed by induction onα_e·E. Consider the morphismp◦fe:P1_→G/P.

LEMMA 3.3. — If the image ofp◦feis a line in the projective embedding given by L then there exists a deformation f′ _∈Hom

α(P1, X)of f not meeting the vertex.

Proof. Indeed, if the image ofp◦feis a line thenf factors through the linear subspace generated by the vertex and this line. It is a Pn+1 _{and the vertex is} a linear subspace of codimension 2. There exists a deformationf′ _{of f in this}

projective space not meeting the vertex. ¤

Let us now assume that the image ofp◦feis not a line, we may apply proposition 2.4 so that there exists a deformation fe′ _{offeand a curve Γ ⊂Xe contracted}

smoothed curve is the image of a morphism fb: P1 _{→ X}e _{of class α. Let usb}

considerf′_=π◦fe′ _andf′′_{=π◦fb. Thenf}′ _{is a deformation off and because}

Γ is contracted byπthe mapf′′_{is a deformation off}′ _{and a fortiori off.}

We have to prove the result onf′′ _{whose lifting isfbof class}

b the dimension of the corresponding component is

Z

e α

c1(T_Xe) + dim(X).

Proof. Theorem 3.1 proves that the set of morphisms f : P1 _{→ X whose}

image does not meet the vertexP_{(V) is a dense open subset of}Hom_α(P1_{, X).}

It is enough to study this open set. Any curve is this open set comes from a unique liftingfe:P1_→Xe _{whose image does not meetE. Letαe∈A1(X}e_{) the}

is thus dominant and birational (the inverse is given by lifting mor-phisms). What is left to prove is that for each αe ∈ ne(α) the image of Hom

e

α(P1,Xe) (which is an irreducible scheme) forms an irreducible

compo-nent of Hom_α(P1_{, X). To prove this it is enough to prove that for any}

In general, suppose there exist f ∈ Hom

e

α(P1,X) such thate f does not

meet the vertex and such that f is the limit of a familly f′

t of morphisms in

Hom

e

α′(P1,Xe). Because the condition of meeting the vertex is closed me may

assume that the elements f′

t do not meet the vertex. In particular projecting

onG/P gives a deformation fromp(f′

t) top(f). This implies that p∗αe=p∗αe′

We begin with the following lemma on root systems:

LEMMA 3.5. — LetGbe a semi-simple Lie group,P ⊂Ga parabolic subgroup,

L a dominant weight in the facet defined byP and R the root lattice, then we have the equivalence

wherec1(G/P)∈Pic(G/P)is considered as a weight and the order is given by the positivity on simple roots.

Proof. Let us first describe c1(G/P) as a weight. Consider the set α(p) of simple root and the latticet(p)∩Q(which is isomorphic to Pic(G/P)) defined in paragraph 1. The latticet(p)∩Qdecomposes into a direct sum of root lattices Ri. Letρi be half the sum of positive roots of the root system corresponding

toRi. Then we have

c1(G/P) = 2X

i

ρi.

IfL≥c1(G/P) then for any simple rootαwe have

hα∨_{, Li ≥ hα}∨_{, c1(G/P)i=}X i

hα∨_{, ρii=}

½

0 if α6∈α(p) 2 if α∈α(p)

and in particular 16∈L(R).

Conversely, suppose that L(R)6= Z_{. BecauseL is in the facet of P we have} hα∨_{, Li = 0 for any simple root α 6∈ α(p). If α is a simple root in α(p)}

then hα∨_{, Li ≥ 2 (otherwise L(R) = Z). We see that for any simple root} hα∨_{, Li ≥ hα}∨_{, c1(G/P)ithusL≥c1(G/P). ¤}

Remark 3.6. — Let αe ∈A1(Xe) such that αe·E ≥0. Recall the notations

β =p∗αe,d=αe·Tp is the relative degree andℓ=αe·p∗L=β·L. Letα=π∗αe considered as an integer. Then the dimension ofHom

e

α(P1,X)e is given by

Z

e α

c1(T_Xe) + dim(Xe) = Z

β

c1(TG/P) +d+ dim(Xe)

= Z

β

c1(TG/P) + (n+ 1)αe·E+nℓ+ dim(Xe)

= β·(c1(TG/P)−L) + (n+ 1)αe·(E+p∗L) + dim(X)e

= β·(c1(TG/P)−L) + (n+ 1)α+ dim(Xe).

So we have the formula

dim(Hom

e

α(P1,Xe)) =

Z

e α

p∗(c1(TG/P)−L) + (n+ 1)α+ dim(Xe).

THEOREM3.7. — AssumeL(R)6=Z_{and letα∈A1(X). Then the irreducible} components ofHom_α(P1_{, X)are indexed by} a

α′≤α

ne(α′_).

Proof. Thanks to lemma 3.2 (this lemma works without the hypothesis L(R) =Z_{) there exists a dense open subset of} Hom_α(P1_{, X) given by}

study this open set. In particular we know that the morphism

is dominant. The classes αe can even be choosen such that Hom

e

α(P1,X) ise

not empty. However the intersectionαe·E need not to be 0. In particular the classesαecan be choosen in

A(α) = a

of the relative degree we have thatHom

e

α′(P1,Xe) is not empty. In particular

e

α′ _{∈ ne(α}′_{) for α}′ _{= π}

∗αe′ = p∗αe·L and we have αe·E =π∗αe−π∗αe′. The

elementα_e is uniquely determined byα_e′ _and

e α·E.

It is enough to prove that the images by π∗ of the irreducible schemes

Hom

contained in the closure of the image ofHom

Remark 3.8. — Let us end with a discussion on the dimensions of the irre-ducible components ofHom_α(P1_{, X)forα∈A1(X).}

(ı) In the first case: L(R) = Z_{, these irreducible components are indexed by} elementsαeinne(α). For such an element we haveαe·E= 0and the dimension of the component is given by

dim(Hom

e

α(P1,Xe)) =

Z

e α

p∗_(c1(T

G/P)−L) + (n+ 1)α+ dim(Xe). The “variable” part in this dimension is the first one and it is given by

β·(c1(TG/P)−L)

withβ =p∗αeand we have α=β·Lso that the “variable” part isβ·c1(TG/P). The elementβ ranges in the subset of the positive cone in the root latticeR(in the projection ofRinPic(G/P)) given by the conditionβ·L=α. In particular if L is not collinear to c1(G/P)the dimensions of the irreducible components are not equal. In this case the varietyHom_α(P1_{, X)is equidimensional if and} only ifL=1

2c1(G/P).

(ıı) In the second case: L(R) 6= Z_{, these irreducible components are indexed} by elements α_e∈ a

α′≤α

ne(α′_{). For such an element we have}

e

α·E ≥0 and the dimension of the component is given by

dim(Hom

e

α(P1,Xe)) =

Z

e α

p∗_(c1(T

G/P)−L) + (n+ 1)α+ dim(Xe). The “variable” part in this dimension is the first one and it is given by

β·(c1(TG/P)−L)

with β = p∗αe. In this case we have β·L =α′ ≤ α. The element β ranges in the subset of the positive cone in the root lattice R (in the projection of

R in Pic(G/P)) given by the condition β·L ≤ α. In particular if L is not collinear toc1(G/P)the dimensions of the irreducible components are not equal (look at the β such that β ·L = α). Furthermore even if L is collinear to

c1(G/P) the dimensions of the irreducible components are not equal unless

L=c1(G/P). In this case the varietyHom_α(P1_{, X)is equidimensional if and} only ifL=c1(G/P).

References

[Fu] William Fulton: Intersection theory. Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 2. Springer-Verlag, Berlin (1998).

[FMcPSS] William Fulton, Robert MacPherson, Frank Sottile and Bernd Sturmfels: Intersection theory on spherical varieties. J. Algebraic Geom. 4 (1995), no. 1.

[FP] William Fulton, Rahul Pandharipande: Notes on stable maps and quantum cohomology. Algebraic geometry—Santa Cruz 1995, 45– 96, Proc. Sympos. Pure Math., 62, Part 2, Amer. Math. Soc., Prov-idence, RI, 1997.

[Gr] Alexander Grothendieck: Techniques de construction et th´eor`emes d’existence en g´eom´etrie alg´ebrique. IV. Les sch´emas de Hilbert. (French). S´eminaire Bourbaki, Vol. 6, Exp. No. 221, 249–276, Soc. Math. France, Paris, (1995).

[HH] Robin Hartshorne and Andr´e Hirschowitz: Smoothing algebraic space curves. Algebraic geometry, Sitges (Barcelona), 98–131, L.N.M., 1124, Springer, Berlin, (1985).

[KP] Bumsiq Kim and Rahul Pandharipande: The connectedness of the moduli space of maps to homogeneous spaces. Symplectic geometry and mirror symmetry (Seoul, 2000), 187–201, World Sci. Publishing, River Edge, NJ, 2001.

[LMS] Venkatramani Lakshmibai, Chitikila Musili and Conjeerveram S. Se-shadri: Geometry of G/P. III. Standard monomial theory for a quasi-minuscule P. Proc. Indian Acad. Sci. Sect. A Math. Sci. 88 (1979), no. 3, 93–177.

[Mo] Shigefumi Mori: Projective manifolds with ample tangent bundles. Ann. of Math. (2) 110 (1979), no. 3, 593–606.

[P1] Nicolas Perrin: Courbes rationnelles sur les vari´et´es homog`enes. Annales de l’Institut Fourier, 52, no.1 (2002), pp 105-132.

[P2] Nicolas Perrin: Rational curves on minuscule Schubert varieties, preprint math.AG/0407123.

[Th] Jesper Funch Thomsen: Irreducibility ofM0,n(G/P, β), Internat. J.

Math. 9 (1998) No. 3. Nicolas Perrin

Institut de Math´ematiques de Jussieu 175 rue du Chevaleret