Rational Curves on

Homogeneous Cones

Nicolas Perrin

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

Communicated by Thomas Peternell

Abstract. A homogeneous cone X 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 morphisms of class α ∈ A_{1}(X) from a rational curve to X.

The situation depends on the line bundle L : if the projectivised tangent 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 ℙ^{1} to any homogeneous cone that is to say a cone X over a homogeneous variety G/P.

This study is motivated by the more general problem of describing the irreducible components of the scheme of morphisms from ℙ^{1} to a variety X endowed with the action of a solvable group with finitely many orbits (for example homogeneous varieties, Schubert varieties or spherical varieties). For homogeneous varieties, this is already known (see [Th], [KP] or [P1]). For Schubert varieties it is not known. We describe in [P2] the irreducible components of this scheme of morphisms when X is a minuscule Schubert variety. The singularities of minuscule Schubert varities are locally isomorphic to cones over homogeneous varieties (see [BP]) so it is natural to address this problem on a general cone over a homogeneous variety. Moreover some non minuscule Schubert varieties are cones over homogeneous varieties. We prove (theorem 0.1) that the situation is more complicated than in the minuscule case but still has a nice description in terms of the combinatorial data of G.

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(H^{0}L). If V is a n-dimensional vector space, it defines a linear subspace ℙ(V) in ℙ(H^{0}L ⊕ V). Let us denote by X = ℭ_{L,n}(G/P) the cone above G/P whose vertex is ℙ(V). Now let U be the open subset of X complementary to Y = ℙ(V). We have a surjective morphism (see paragraph 1):

s : Pic(U)^{∨} → A_{1}(X).

For any class α ∈ A_{1}(X), we can consider the following morphism:

i : ∐_{s(β)=α} Hom_{β}(ℙ^{1}, U) → Hom_{α}(ℙ^{1}, X)

where Hom_{α}(ℙ^{1}, X) is the scheme of morhisms f : ℙ^{1} → X with f_{∗}[ℙ^{1}] = α and Hom_{β}(ℙ^{1}, U) is the scheme of morhisms g : ℙ^{1} → U such that [g] = β where [g] is the linear function L ↦ deg(g^{∗}L) on Pic(U). As Y = 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 where i is dominant. In this case we may apply the results of [P1] to prove that Hom_{β}(ℙ^{1}, U) is irreducible as soon as it is non empty and the images of the irreducible varieties Hom_{β}(ℙ^{1}, U) will give the irreducible components of Hom_{α}(ℙ^{1}, X). The expected components are thus indexed by the subset 𝔫𝔢(α) of Pic(U)^{∨} given by elements β such that s(β) = α and Hom_{β}(ℙ^{1}, U) is non empty.

This set can be discribed in terms of roots: the ample divisor L is a dominant weight in the facet of the parabolic P. An element α ∈ A_{1}(X) is completely determined by α · L = d ∈ ℤ. Denote by 𝔫𝔢_{B}(α) the set of all elements β in the cone generated by the positive roots such that ⟨β^{∨}, L⟩ = d. This is a subset of A_{1}(G/B). Then 𝔫𝔢(α) is its image in A_{1}(G/P) (see paragraph 1 for a more details). We prove the

THEOREM 0.1. — Let R be the root lattice. (ı) If L(R) = ℤ, then the irreducible components of the scheme of morphisms Hom_{α}(ℙ^{1}, X) are indexed by 𝔫𝔢(α). (ıı) If L(R) ≠ ℤ (i.e. if we have L > c_{1}(T_{G/P})), then the irreducible components of the scheme Hom_{α}(ℙ^{1}, ℭ(G/P)) are indexed by ∐_{α′≤α} 𝔫𝔢(α′).

We will see (paragraph 1) that A_{1}(X) ≃ ℤ so that α′ ≤ α in A_{1}(X) means that the same inequality holds in ℤ. In the second case, a general curve can meet the vertex of the cone. The integer α − 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 generaly 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) If P = B is a Borel subgroup and if we choose for L the Plücker embedding (or equivalently L = ρ as a weight where ρ is 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 algebra 𝔤̂ (see paragraph 1 for the general case).

Here is an outline of the paper. In the first paragraph we define the surjective map s of the introduction and the set 𝔫𝔢(α) for a homogeneous cone X. In the second paragraph we study the scheme of morphisms from ℙ^{1} to the blowing-up X̃ of the cone X 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 morphism f : ℙ^{1} → X be factorised in U (modulo deformation). We do this by lifting f in f̃ on X̃ and the problem becomes: does the lifted curve f̃ of a general curve f meet the exceptional divisor E. If it is the case then we add a “line” Γ ⊂ E (this is possible only when L(R) = ℤ) with Γ·E = −1 and smooth the union f̃(ℙ^{1})∪Γ. The intersection with E is lowered by one in the operation. We conclude by induction on the number of intersection of f̃ with E.

We end with a discussion on the dimensions of the components, in particular the variety Hom_{α}(ℙ^{1}, X) is equidimensional if and only if L = \frac{1}{2}c_{1}(G/P) or L = c_{1}(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)^{∨} → A_{1}(X) and define the set of classes 𝔫𝔢(α) for α ∈ A_{1}(X).

If X is a scheme of dimension n, we denote by Z_{∗}(X) the group of 1-cycles on X and by Z_{∗}^{≡}(X) and Z_{∗}^{r}(X) the subgroups of cycles trivial for the numerical and rational equivalence. Let us denote by N_{∗}(X) and A_{∗}(X) the corresponding quotients. The Picard group is the image in A_{n−1}(X) of the subgroup of Cartier divisors in Z_{n−1}(X).

LEMMA 1.1. — Let X = ℭ_{L,n}(G/P) be a cone over a homogeneous variety G/P then (ı) Pic(X) ≃ N^{1}(X), (ıı) A_{1}(X) ≃ N_{1}(X). In particular we have A_{1}(X) ≃ Pic(X)^{∨} and they are isomorphic to ℤ.

Proof. Consider the decomposition V ⊕ H^{0}L. The following group

G′ = ( \begin{matrix}GL(V) & Hom(H^{0}L, V) \\ 0 & G \end{matrix} )

acts on X and the unipotent part U(G′) acts on X with finitely many orbits. Remark that it is not the case if we only take the unipotent part U of GL(V)×G. Indeed, take for example V = ℂ and G = GL(V) = GL_{1}(ℂ). Then G/P is a point and X is a projective line but U is the trivial group and has infinitely many orbits in X. The same problem appears in the general case. (ı) Thanks to the results of [FMcPSS] the groups A_{∗}(X) are free generated by invariant subvarieties. The Picard group is contained in A_{n−1}(X) and is in particular free. Thanks to [Fu] Example 19.3.3. this implies that Pic(X) ≃ N^{1}(X). (ıı) The results of [FMcPSS] also imply that A_{1}(X) is generated by the onedimensional invariant subvarieties. The only such subvariety is

• the fibre of the cone over the 0-dimensional orbit in G/P if dim V = 1;

• the 1 dimensional orbit in ℙ(V) if dim V ≥ 2.

We get the isomorphism A_{1}(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 between N_{1}(X) and N^{1}(X). □

Let U be the smooth locus of X, it is also the dense orbit under G′ in X. 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) = A_{n−1}(U) ≃ A_{n−1}(X). We now have the following inclusion:

Pic(X) ⊂ A_{n−1}(X) ≃ Pic(U)

giving the surjection

s : Pic(U)^{∨} → A_{1}(X).

With these notations we make the following:

DEFINITION 1.2. — Let α ∈ A_{1}(X). We define the set 𝔫𝔢(α) ⊂ A_{n−1}(X)^{∨}. Let us make the identification A_{n−1}(X) ≃ Pic(U). The elements of 𝔫𝔢(α) are the elements β ∈ Pic(U)^{∨} such that s(β) = α and 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 fix T a maximal Torus in G, fix B a Borel subgroup containing T and suppose that B ⊂ P. Let us denote by Δ the set of all roots, by Δ^{+} (resp. Δ^{−}) the set of positive (resp. negative) roots and by S the set of simple roots associated to the data (G, T, B). Denote by 𝔤, 𝔱 and 𝔭 the Lie algebras of G, T and P and define

α(𝔭) = {α ∈ S / 𝔤_{α} ⊂ 𝔭 and 𝔤_{−α} ⊄ 𝔭}.

Now set 𝔱(𝔭)^{∗} as the subvector space of 𝔱^{∗} generated by the roots in α(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 of X in Pic(U) ≃ Pic(G/P) ≃ 𝔱(𝔭) ∩ Q is given by the the line generated by λ (the weight associated to L). 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 cone 𝔱^{∗}/𝔱(𝔭)^{∗} ∩ R^{+} (see [P1]). Then we have

𝔫𝔢(α) = {β ∈ 𝔱^{∗}/𝔱(𝔭)^{∗} ∩ R^{+} / ⟨β^{∨}, λ⟩ = α · L}

where the integer ⟨β^{∨}, λ⟩ is well defined because λ ∈ 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) ∩ Q is a direct sum of weight lattices of semi-simple Lie algebras (𝔤_{i})_{i∈[1,r]}. We just have to take

λ = ∑_{i∈[1,r]} ρ_{i}

where ρ_{i} is half the sum of positive roots in 𝔤_{i}. Let us denote by 𝔦𝔯𝔤̂_{i}(ℓ) the set of isomorphism classes of irreductible integrable representations of level exactely ℓ of the affine Lie algebra 𝔤̂_{i}. Then we have

𝔫𝔢(α) = ∏_{ℓ_{1}+···+ℓ_{r}=α·L} 𝔦𝔯𝔤̂_{i}(ℓ_{i}).

In particular if P = B is a Borel subgroup of G then r = 1 and G_{1} = G and we recover the example of the introduction:

𝔫𝔢(α) = 𝔦𝔯𝔤̂(α · L).

Remark 1.4. — (ı) The scheme Hom_{α}(ℙ^{1}, X) is the scheme of morphisms from ℙ^{1} to X of class α (for more details see [Gr] and [Mo]). In general, this will just mean that α ∈ A_{1}(X) and that f_{∗}[ℙ^{1}] = α but sometimes (in particular in the introduction for the open part U) we consider α ∈ Pic(X)^{∨} and the class of a morphism f : ℙ^{1} → X will be the linear form Pic(X) → ℤ given by L ↦ 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 ∈ A_{1}(X) and F a vector bundle on X we will denote α · F = ∫_{α} c_{1}(F) by abuse of notation.

2 Resolution

Recall that we denote by X the cone ℭ_{L,n}(G/P). Let X̃ be the blowing-up of X in ℙ(V). It is smooth and isomorphic to

ℙ_{G/P} ((V ⊗ O_{G/P}) ⊕ L).

Let us denote by p the projection from X̃ to G/P and by π : X̃ → X the blowing-up. The morphism p has natural sections given by points of ℙ(V) or equivalently by surjective morphisms L ⊕ (V ⊗ O_{G/P}) → V ⊗ O_{G/P} → O_{G/P} .

2.1 Cycles on X̃

LEMMA 2.1. — (ı) Rational and numerical equivalences coincide on X̃. In particular we have A_{1}(X̃) ≃ Pic(X̃)^{∨} ≃ A_{n−1}(X̃)^{∨}. (ıı) We have Pic(X̃) ≃ Pic(G/P)⊕ℤ with the factor ℤ generated by a p-relative ample class.

Proof. (ı) Rational and numerical equivalence coincide on G/P. Moreover the fibration in projetive spaces X̃ → G/P has sections so that rational and numerical equivalences coincide on X̃. This in particular implies that Pic(X̃) = A_{n−1}(X̃) = N^{1}(X̃) and A_{1}(X̃) = N_{1}(X̃) and the duality follows. (ıı) The variety X̃ is a ℙ^{n}-bundle over G/P with sections so we get that

Pic(X̃) ≃ Pic(G/P) ⊕ ℤ

with the factor ℤ generated by a p-relative ample class, the relative tangent sheaf T_{p} of p is n + 1 times this class on the factor ℤ. □

Any element α̃ ∈ A_{1}(X̃) ≃ Pic(X̃)^{∨} is given by the class β = p_{∗}α̃ ∈ A_{1}(G/P) and the relative degree d = α̃ · T_{p}. We will use the notation ℓ = β · L = α̃ · p^{∗}L. Let us denote by E the exceptional divisor on X̃, it is a trivial ℙ^{n−1} bundle over G/P given by the surjection L⊕(V ⊗O_{G/P}) → V ⊗O_{G/P}. Then we have:

α̃ · E = \frac{d − nℓ}{n + 1},

it has to be an integer so that d ≡ nℓ mod n + 1.

Let us consider the following morphism still denoted by p:

p : Homα̃(ℙ^{1}, X̃) → Hom_{β}(ℙ^{1}, G/P).

PROPOSITION 2.2. — Thanks to the morphism p, the scheme Homα̃(ℙ^{1}, X̃) is an open subset of a projective bundle over Hom_{β}(ℙ^{1}, G/P).

Proof. This generalises proposition 4 of [P1] in the case where the relative degree α̃ · T_{p} is negative. This is possible because the vector bundle associated to the ℙ^{n} fibration has a decomposition L ⊕ (V ⊗ O_{G/P}). We only describe the fibers, for the structure of projective bundle see [P1] proposition 4. Let f : ℙ^{1} → G/P, we have to calculate the fiber of p above f. The fiber is given by sections of the ℙ^{n}-bundle f^{∗}(p) : ℙ_{ℙ^{1}}((V ⊗O_{ℙ^{1}})⊕O_{ℙ^{1}}(ℓ)) → ℙ^{1} whose relative degree is d. In other words the fiber is given by surjections (V ⊗O_{ℙ^{1}})⊕ O_{ℙ^{1}}(ℓ) → O_{ℙ^{1}}(x) modulo scalar multiplication where d = (n+1)x−ℓ. The fiber is therefore isomorphic to an open subset of ℙ(Hom((V ⊗O_{ℙ^{1}})⊕O_{ℙ^{1}}(ℓ), O_{ℙ^{1}}(x)). Let us remark that if Hom_{β}(ℙ^{1}, G/P) is not empty then we have ℓ ≥ 0 and in this case Homα̃(ℙ^{1}, X̃) is not empty if and only if x ≥ 0 when n ≥ 2 and if and only if x = 0 or x ≥ ℓ when n = 1. In terms of d this means that d = −ℓ or d ≥ nℓ if n = 1 and d ≥ −ℓ if n ≥ 2. In any cases, if Homα̃(ℙ^{1}, X̃) is not empty then x ≥ 0. There are two cases:

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

\frac{n}{n + 1}(ℓ + d) + n − 1.

• If x ≥ ℓ then the fiber is of dimension d + n. □

Let α̃ ∈ A_{1}(X̃) such that Homα̃(ℙ^{1}, X̃) is not empty. This is equivalent to the fact that β ∈ A_{1}(G/P) is positive (see [P1], it is equivalent to the fact that Hom_{β}(ℙ^{1}, G/P) is non empty) and such that d = −ℓ or d ≥ nℓ if n = 1, d ≥ −ℓ if n ≥ 2 (recall that ℓ = β · L).

COROLLARY 2.3. — The scheme Homα̃(ℙ^{1}, X̃) is irreducible of dimension

• ∫α̃ c_{1}(TX̃) + dim(X̃) if d ≥ nℓ

• ∫α̃ c_{1}(TX̃) + dim(X̃) − α̃ · E − 1 if d < nℓ.

Proof. We just use the preceding proposition and the fact proved in [P1] that the scheme Hom_{β}(ℙ^{1}, G/P) is irreductible of dimension ∫_{β} c_{1}(T_{G/P}) + dim(G/P). Remark that in the last case we have nℓ > d so that the dimension of Homα̃(ℙ^{1}, X̃) is still greater than the expected dimension ∫α̃ c_{1}(TX̃) + dim(X̃). □

2.2 Smoothing curves on X̃

In this paragraph we will prove some results on curves on X̃.

PROPOSITION 2.4. — Assume that L(R) = ℤ. Let α̃ ∈ A_{1}(X̃), f̃ ∈ Homα̃(ℙ^{1}, X̃) such that f̃(ℙ^{1}) ⊄ E and α̃·E > 0. Assume that the image of p ∘ f̃: ℙ^{1} → G/P is not a line in the embedding given by L. Then there exists a deformation f̃′ of f̃ and a curve Γ ⊂ X̃ contracted by π with Γ · E = −1 such that the curve f̃′(ℙ^{1}) ∪ Γ can be smoothed. The smoothed curve is the image of a morphism f̂: ℙ^{1} → X̃.

Proof. Let (x, v) ∈ E ≃ G/P ×ℙ(V) be a point in the intersection f̃(ℙ^{1})∩E. Let us first remark that we may assume (after deformation) that f̃(ℙ^{1}) is a nodal curve. Indeed, because p ∘ f̃: ℙ^{1} → G/P is not a line, this implies in particular that G/P is not ℙ^{1} so it is of dimension at least 2. The results of [P1] prove that a general curve in G/P is nodal and so is f̃(ℙ^{1}) if f̃ is general.

LEMMA 2.5. — There exists a deformation f̃′ of f̃ and a rational curve Γ in X̃ such that [Γ] · E = −1, [Γ] · L = 1 and meeting f̃′(ℙ^{1}) in exactly one point.

Proof. Let us consider the lines in G/P that is to say the rational curves Γ′ in G/P such that [Γ′]·L = 1. Such curves exist because we have L(R) = ℤ. Let Γ′ be such a line passing through p(x, v) = x ∈ G/P and let Γ be the section of Γ′ in E given by the point v ∈ ℙ(V). This curve is contracted by π to the point v ∈ ℙ(V), its intersection with E is given by −[Γ′] · L = −1. As we assumed that p ∘ f̃(ℙ^{1}) is not a line then Γ′ meets p ∘ f̃(ℙ^{1}) in a finite number of points: x and other points (x_{i}). The morphism f̃ is given by a section of the projective bundle over p ∘ f̃ that is to say by a surjection

s : (V ⊗ O_{ℙ^{1}}) ⊕ O_{ℙ^{1}}(ℓ) → O_{ℙ^{1}} (\frac{d + ℓ}{n + 1}) .

To deform f̃we can deform this surjection s in s′ such that at x, we have s′_{x} = s_{x} and at x_{i} we have s′_{x_{i}} ≠ s_{x_{i}} for all i. This gives the required deformation. □

LEMMA 2.6. — The curve f̃′(ℙ^{1}) ∪ Γ can be smoothed. The smoothed curve is the image of a morphism f̂: ℙ^{1} → X̃ of class α̂ with

α̂ · (p^{∗}L + E) = α̃ · (p^{∗}L + E) and α̂ · E < α̃ · E.

Proof. If the smoothing exists then we have α̂ = α̃ + [Γ] so α̂ · E = α̃ · E − 1. Furthermore we have p^{∗}L + E = π^{∗}L so that

α̂ · (p^{∗}L + E) = α̂ · π^{∗}L = π_{∗}α̂ · L = π_{∗}α̃ · L = α̃ · π^{∗}L = α̃ · (p^{∗}L + E).

This simply comes from the fact that π_{∗}[Γ] = 0. Let us note that the curves f′ = π ∘ f̃ and f′′ = π ∘ f̂ have the same degrees but the curve f′′ meets the vertex in one point less than f′.

To smooth f̃′(ℙ^{1}) ∪ Γ we use the following result proved in [HH] for ℙ^{3} but valid for any smooth projective variety:

THEOREM 2.7. — Let Z be a smooth projective variety and let C be a nodal curve in Z. Assume that the cohomology group H^{1}T_{Z}|_{C} is trivial then C can be smoothed.

As f̃(ℙ^{1}) is nodal and thanks to the previous lemma we know that f̃′(ℙ^{1}) ∪ Γ is nodal. We just have to prove that the cohomology group H^{1}(TX̃|_{f̃′(ℙ^{1})∪Γ}) is trivial. We have the exact sequences

0 → T_{p} → TX̃ → p^{∗}T_{G/P} → 0 and 0 → O_{f̃′(ℙ^{1})}(−Q) → O_{f̃′(ℙ^{1})∪Γ} → O_{Γ} → 0

where Q is the intersection point of f̃′(ℙ^{1}) and Γ. We just have to prove the vanishing of the following cohomology groups:

H^{1}(p^{∗}T_{G/P} |_{Γ}) ; H^{1}(p^{∗}T_{G/P} |_{f̃′(ℙ^{1})}(−Q)) ; H^{1}(T_{p}|_{Γ}) and H^{1}(T_{p}|_{f̃′(ℙ^{1})}(−Q)).

The first two groups are respectively equal to H^{1}(T_{G/P} |_{Γ′}) and H^{1}(T_{G/P} |_{p(f̃′(ℙ^{1}))}(−Q)) where we denoted Γ′ = p(Γ). They are trivial

because T_{G/P} is globally generated and Γ′ and p(f̃′(ℙ^{1})) are rational curves. Let us denote by O_{p}(1) the tautological quotient of the projective bundle associated to (V ⊗O_{X})⊕L, the relative tangent sheaf is given by T_{p} = Coker(OX̃ → ((V^{∨} ⊗ OX̃) ⊕ L^{∨}) ⊗ O_{p}(1)). In particular we have:

T_{p}|_{Γ} = Coker(O_{ℙ^{1}} → (V^{∨} ⊗ O_{ℙ^{1}}) ⊕ O_{ℙ^{1}}(−1)) and

T_{p}|_{f̃′(ℙ^{1})} = Coker (O_{ℙ^{1}} → (V^{∨} ⊗ O_{ℙ^{1}} (\frac{d + ℓ}{n + 1})) ⊕ O_{ℙ^{1}} (\frac{d − nℓ}{n + 1})) .

This proves that the group H^{1}(T_{π}|_{Γ}) vanishes. Furthermore, since f̃ exists we must have \frac{d+ℓ}{n+1} ≥ 0 (see proposition 2.2) and \frac{d−nℓ}{n+1} = α̃ · E > 0 so that H^{1}(T_{p}|_{f̃′(ℙ^{1})}(−Q)) also vanishes. □

3 Homogeneous cones

Recall that we denote by X the cone ℭ_{L,n}(G/P). In this paragraph we study the irreducible components of the scheme Hom_{α}(ℙ^{1}, X) where α ∈ A_{1}(X). Recall that A_{1}(X) ≃ ℤ and under this identification α is just the degree of the corresponding curve.

3.1 The case L(R) = ℤ

THEOREM 3.1. — Assume that L(R) = ℤ, let α ∈ A_{1}(X) and f ∈ Hom_{β}(ℙ^{1}, X). Then there exists a deformation f′ of f such that f′ 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}(ℙ^{1}, X) such that f factors through the vertex ℙ(V) of the cone. Then there exists a deformation f′ of f in Hom_{α}(ℙ^{1}, X) such that f′(ℙ^{1}) does not factor through the vertex.

Proof. Let x ∈ G/P and consider the linear subspace generated by x and ℙ(V). It is a projective space contained in X containing ℙ(V) as a hyperplane and containing f(ℙ^{1}). In this projective space we can deform the morphism f so that it does not factor through ℙ(V) any more. □

A general morphism f ∈ Hom_{α}(ℙ^{1}, X) does not factor through the vertex ℙ(V) of the cone so it can be lifted in a morphism f̃: ℙ^{1} → X̃. Let α̃ ∈ A_{1}(X̃) the class of f̃, we have π_{∗}α̃ = α. Because f does not factor through the vertex, the morphism f̃ does not factor through the exceptional divisor E so we have: α̃ · E ≥ 0. If α̃ · E = 0, then f̃(ℙ^{1}) does not meet E thus f does not meet the vertex and we are done. Let us assume that α̃ · E > 0. We proceed by induction on α̃ · E. Consider the morphism p ∘ f̃: ℙ^{1} → G/P.

LEMMA 3.3. — If the image of p∘f̃ is a line in the projective embedding given by L then there exists a deformation f′ ∈ Hom_{α}(ℙ^{1}, X) of f not meeting the vertex.

Proof. Indeed, if the image of p ∘ f̃ is a line then f factors through the linear subspace generated by the vertex and this line. It is a ℙ^{n+1} and the vertex is a linear subspace of codimension 2. There exists a deformation f′ of f in this projective space not meeting the vertex. □

Let us now assume that the image of p∘f̃is not a line, we may apply proposition 2.4 so that there exists a deformation f̃′ of f̃ and a curve Γ ⊂ X̃ contracted by π with Γ · E = −1 such that the curve f̃′(ℙ^{1}) ∪ Γ can be smoothed. The smoothed curve is the image of a morphism f̂ : ℙ^{1} → X̃ of class α̂. Let us consider f′ = π ∘ f̃′ and f′′ = π ∘ f̂. Then f′ is a deformation of f and because Γ is contracted by π the map f′′ is a deformation of f′ and a fortiori of f. We have to prove the result on f′′ whose lifting is f̂ of class α̂. But we have α̂ · E = α̃ · E − 1 so the result is true by induction. □

THEOREM 3.4. — Assume L(R) = ℤ and let α ∈ A_{1}(X) then the irreducible components of the scheme Hom_{α}(ℙ^{1}, X) are indexed by 𝔫𝔢(α). For α̃ ∈ 𝔫𝔢(α) the dimension of the corresponding component is

∫α̃ c_{1}(TX̃) + dim(X).

Proof. Theorem 3.1 proves that the set of morphisms f : ℙ^{1} → X whose image does not meet the vertex ℙ(V) is a dense open subset of Hom_{α}(ℙ^{1}, X). It is enough to study this open set. Any curve is this open set comes from a unique lifting f̃: ℙ^{1} → X̃ whose image does not meet E. Let α̃ ∈ A_{1}(X̃) the class of f̃, since α̃ · E = 0 we have α̃ ∈ Pic(U)^{∨} and in fact α̃ ∈ 𝔫𝔢(α). The morphism

π_{∗} : ∐_{α̃∈𝔫𝔢(α)} Homα̃(ℙ^{1}, X̃) → Hom_{α}(ℙ^{1}, X)

is thus dominant and birational (the inverse is given by lifting morphisms). What is left to prove is that for each α̃ ∈ 𝔫𝔢(α) the image of Homα̃(ℙ^{1}, X̃) (which is an irreducible scheme) forms an irreducible component of Hom_{α}(ℙ^{1}, X). To prove this it is enough to prove that for any α̃ and α′ in 𝔫𝔢(α) the image of Homα̃(ℙ^{1}, X̃) is not contained in the closure of Hom_{α′}(ℙ^{1}, X̃) in Hom_{α}(ℙ^{1}, X). This would be trivial if the scheme ∐_{α̃∈𝔫𝔢(α)} Homα̃(ℙ^{1}, X̃) was equidimensional (it is the case if L = \frac{1}{2}c_{1}(G/P)). In general, suppose there exist f ∈ Homα̃(ℙ^{1}, X̃) such that f does not meet the vertex and such that f is the limit of a familly f′_{t} of morphisms in Hom_{α′}(ℙ^{1}, X̃). 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 on G/P gives a deformation from p(f′_{t}) to p(f). This implies that p_{∗}α̃ = p_{∗}α′ but as α̃ · E = 0 = α′ · E we have α̃ = α′. The dimension comes from corollary 2.3. □

3.2 The case L(R) ≠ ℤ

We begin with the following lemma on root systems:

LEMMA 3.5. — Let G be a semi-simple Lie group, P ⊂ G a parabolic subgroup, L a dominant weight in the facet defined by P and R the root lattice, then we have the equivalence

L(R) ≠ ℤ ⟺ L ≥ c_{1}(G/P)

where c_{1}(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 c_{1}(G/P) as a weight. Consider the set α(𝔭) of simple root and the lattice 𝔱(𝔭)∩Q (which is isomorphic to Pic(G/P)) defined in paragraph 1. The lattice 𝔱(𝔭) ∩ Q decomposes into a direct sum of root lattices R_{i}. Let ρ_{i} be half the sum of positive roots of the root system corresponding to R_{i}. Then we have

c_{1}(G/P) = 2 ∑_{i} ρ_{i}.

If L ≥ c_{1}(G/P) then for any simple root α we have

⟨α^{∨}, L⟩ ≥ ⟨α^{∨}, c_{1}(G/P)⟩ = ∑_{i} ⟨α^{∨}, ρ_{i}⟩ = \{ \begin{matrix}0 & if α ∉ α(𝔭) \\ 2 & if α ∈ α(𝔭) \end{matrix}

and in particular 1 ∉ L(R). Conversely, suppose that L(R) ≠ ℤ. Because L is in the facet of P we have ⟨α^{∨}, L⟩ = 0 for any simple root α ∉ α(𝔭). If α is a simple root in α(𝔭) then ⟨α^{∨}, L⟩ ≥ 2 (otherwise L(R) = ℤ). We see that for any simple root ⟨α^{∨}, L⟩ ≥ ⟨α^{∨}, c_{1}(G/P)⟩ thus L ≥ c_{1}(G/P). □

Remark 3.6. — Let α̃ ∈ A_{1}(X̃) such that α̃ · E ≥ 0. Recall the notations β = p_{∗}α̃, d = α̃ · T_{p} is the relative degree and ℓ = α̃ · p^{∗}L = β · L. Let α = π_{∗}α̃ considered as an integer. Then the dimension of Homα̃(ℙ^{1}, X̃) is given by

∫α̃ c_{1}(TX̃) + dim(X̃) = ∫_{β} c_{1}(T_{G/P}) + d + dim(X̃)
= ∫_{β} c_{1}(T_{G/P}) + (n + 1)α̃ · E + nℓ + dim(X̃)
= β · (c_{1}(T_{G/P}) − L) + (n + 1)α̃ · (E + p^{∗}L) + dim(X̃)
= β · (c_{1}(T_{G/P}) − L) + (n + 1)α + dim(X̃).

So we have the formula

dim(Homα̃(ℙ^{1}, X̃)) = ∫α̃ p^{∗}(c_{1}(T_{G/P}) − L) + (n + 1)α + dim(X̃).

THEOREM 3.7. — Assume L(R) ≠ ℤ and let α ∈ A_{1}(X). Then the irreducible components of Hom_{α}(ℙ^{1}, X) are indexed by ∐_{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}(ℙ^{1}, X) given by morphisms f that do not factor through the vertex of the cone. It is enough to study this open set. In particular we know that the morphism

π_{∗} : ∐_{α̃∈A_{1}(X̃), π_{∗}α̃=α} Homα̃(ℙ^{1}, X̃) → Hom_{α}(ℙ^{1}, X)

is dominant. The classes α̃ can even be choosen such that Homα̃(ℙ^{1}, X̃) is not empty. However the intersection α̃ · E need not to be 0. In particular the classes α̃ can be choosen in

A(α) = ∐_{α′≤α} 𝔫𝔢(α′)

where α′ = p_{∗}α̃ · L and α − α′ = α̃ · E. Indeed let α̃ ∈ A_{1}(X̃) and set as usual β = p_{∗}α̃. Then there exists a unique element α′ ∈ A_{1}(X̃) such that p_{∗}α′ = β and α′ · E = 0 (take nβ · L for the relative degree). If α̃ is such that Homα̃(ℙ^{1}, X̃) is not empty then β is effective and because of the value of the relative degree we have that Hom_{α′}(ℙ^{1}, X̃) is not empty. In particular α′ ∈ 𝔫𝔢(α′) for α′ = π_{∗}α′ = p_{∗}α̃ · L and we have α̃ · E = π_{∗}α̃ − π_{∗}α′. The element α̃ is uniquely determined by α′ and α̃ · E. It is enough to prove that the images by π_{∗} of the irreducible schemes Homα̃(ℙ^{1}, X̃) for α̃ ∈ A(α) are the irreducible components. In other words we have to prove that for any α̃ and α̂ in A(α) the image of Homα̃(ℙ^{1}, X̃) is not contained in the closure of the image of Homα̂(ℙ^{1}, X̃) in Hom_{α}(ℙ^{1}, X). Let f̃ ∈ Homα̃(ℙ^{1}, X̃) a generic point and f̂_{t} a familly of morphisms in Homα̂(ℙ^{1}, X̃) such that π ∘ f̂_{t} converges to π ∘ f̃. In the compactification of Homα̂(ℙ^{1}, X̃) by stable maps (see for example [FP]), the familly f̂_{t} has a limit say f̂which is a morphism from a tree ∪_{i}D_{i} of rational curves to X̃. Then we must have π ∘ f̂ = π ∘ f̃ as stable maps. In particular all but one of the images by f̂ of the irreducible components of the tree are contracted by π. To fix notation say that D_{i} is contracted by π for i ≥ 2 and π ∘ f̂|_{D_{1}} = π ∘ f̃. Because f̃ is generic, it is not contained in the exceptional divisor so that the equality π ∘ f̂|_{D_{1}} = π ∘ f̃ implies that f̂|_{D_{1}} = f̃. We see that f̂_{∗}[D_{1}] = α̃ so that

α̂ = f̂_{∗}[D_{1}] + ∑_{i≥2} f̂_{∗}[D_{i}] = α̃ + ∑_{i≥2} f̂_{∗}[D_{i}].

In particular we have β̂ = p_{∗}α̂ ≥ p_{∗}α̃ = β̃ and because L(R) ≠ ℤ we know thanks to lemma 3.5 that L ≥ c_{1}(G/P) and we get

β̂ · (c_{1}(T_{G/P}) − L) ≤ β̃ · (c_{1}(T_{G/P}) − L).

As α = π_{∗}α̂ = π_{∗}α̃ we see that

dim(Homα̂(ℙ^{1}, X̃)) ≤ dim(Homα̃(ℙ^{1}, X̃)).

But the morphism π_{∗} is generically injective on Homα̂(ℙ^{1}, X̃) and Homα̃(ℙ^{1}, X̃) so that the scheme π_{∗}(Homα̃(ℙ^{1}, X̃)) cannot be in the closure of π_{∗}(Homα̂(ℙ^{1}, X̃)). □

Remark 3.8. — Let us end with a discussion on the dimensions of the irreducible components of Hom_{α}(ℙ^{1}, X) for α ∈ A_{1}(X). (ı) In the first case: L(R) = ℤ, these irreducible components are indexed by elements α̃ in 𝔫𝔢(α). For such an element we have α̃·E = 0 and the dimension of the component is given by

dim(Homα̃(ℙ^{1}, X̃)) = ∫α̃ p^{∗}(c_{1}(T_{G/P}) − L) + (n + 1)α + dim(X̃).

The “variable” part in this dimension is the first one and it is given by

β · (c_{1}(T_{G/P}) − L)

with β = p_{∗}α̃ and we have α = β · L so that the “variable” part is β · c_{1}(T_{G/P}). 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 to c_{1}(G/P) the dimensions of the irreducible components are not equal. In this case the variety Hom_{α}(ℙ^{1}, X) is equidimensional if and

only if L = \frac{1}{2}c_{1}(G/P).

(ıı) In the second case: L(R) ≠ ℤ, these irreducible components are indexed by elements α̃ ∈ ∐_{α′≤α} 𝔫𝔢(α′). For such an element we have α̃ · E ≥ 0 and the

dimension of the component is given by

dim(Homα̃(ℙ^{1}, X̃)) = ∫α̃ p^{∗}(c_{1}(T_{G/P}) − L) + (n + 1)α + dim(X̃).

The “variable” part in this dimension is the first one and it is given by

β · (c_{1}(T_{G/P}) − L)

with β = p_{∗}α̃. 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 to c_{1}(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 c_{1}(G/P) the dimensions of the irreducible components are not equal unless L = c_{1}(G/P). In this case the variety Hom_{α}(ℙ^{1}, X) is equidimensional if and only if L = c_{1}(G/P).

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Nicolas Perrin

Institut de Mathématiques de Jussieu 175 rue du Chevaleret 75013 Paris, France nperrin@math.jussieu.fr 638

Documenta Mathematica 9 (2004)