ARE MINIMAL DEGREE RATIONAL CURVES DETERMINED BY THEIR TANGENT VECTORS?

STEFAN KEBEKUS AND SÁNDOR J. KOVÁCS

Abstract. Let X be a projective variety which is covered by rational curves, for instance a Fano manifold over the complex numbers. In this setup, characterization and classification problems lead to the natural question: “Given two points on X, how many minimal degree rational curve are there which contain those points?”. A recent answer to this question led to a number of new results in classification theory. As an infinitesimal analogue, we ask “How many minimal degree rational curves exist which contain a prescribed tangent vector?”

In this paper, we give sufficient conditions which guarantee that every tangent vector at a general point of X is contained in at most one rational curve minimal degree. As an immediate application, we obtain irreducibility criteria for the space of minimal rational curves.

Contents

1. Introduction 1 2. ℙ^{1}-bundles with double sections 3 3. Dubbies 7 4. Proofs of the Main Theorems 13 5. Applications 17 References 19

1. Introduction

The study of rational curves of minimal degree has proven to be a very useful tool in Fano geometry. The spectrum of application covers diverse topics such as deformation rigidity, stability of the tangent sheaf, classification problems or the existence of non-trivial finite morphisms between Fano manifolds; see [Hwa01] for an overview.

In this paper we will consider the situation where X is a projective variety, which is covered by rational curves, e.g. a Fano manifold over ℂ. An example of that is ℙ^{n}, which is covered by lines. The key point of many applications of minimal degree rational curves is showing that the curves in question are similar to lines in certain respects. For instance, one may ask:

Date: October 17, 2002. 1991 Mathematics Subject Classification. 14M99, 14J45, 14J99. Key words and phrases. Fano manifold, rational curve of minimal degree. S. Kebekus gratefully acknowledges support by the Forschungsschwerpunkt “Globale Methoden in der kom-

plexen Geometrie” of the Deutsche Forschungsgemeinschaft. S. Kovács was supported in part by NSF Grants DMS-0196072 and DMS-0092165 and by a Sloan Research Fellowship.

Question 1.1. Under what conditions does there exist a unique minimal degree rational curve containing two given points?

This question found a sharp answer in [Keb02a], see [CMSB00] and [Keb02b] for a number of applications. The argument used there is based on a criterion of Miyaoka, who was the first to observe that if the answer to the question is “No”, then a lot of minimal degree curves are singular. We refer to [Kol96, Prop. V.3.7.5] for a precise statement.

As an infinitesimal analogue of this question one may ask the following:

Question 1.2. Are there natural conditions that guarantee that a minimal degree rational curve is uniquely determined by a tangent vector?

Although a definite answer to the latter question would be as interesting as one to the former, it seems that Question 1.2 has hardly been studied before. This paper is a first attempt in that direction. We give a criterion which parallels Miyaoka’s approach.

Theorem 1.3. Let X be a projective variety over an algebraically closed field k and H ⊂ RatCurves^{n}(X) a proper, covering family of rational curves such that none of the associated curves has a cuspidal singularity. If char(k) ≠ 0, assume additionally that there exists an ample line bundle L ∈ Pic(X) such that for every ℓ ∈ H the intersection number L.ℓ of L is coprime to char(k).

Then, if x ∈ X is a general point, all curves associated with the closed subfamily

H_{x} := {ℓ ∈ H | x ∈ ℓ} ⊂ H

are smooth at x and no two of them share a common tangent direction at x.

Remark 1.3.1. In Theorem 1.3 we do not assume that H is irreducible or connected. That will later be important for the applications.

Remark 1.3.2. We refer the reader to Chapter 3.3.1 for a brief review of the space RatCurves^{n}(X) of rational curves. The volume [Kol96] contains a thorough discussion.

If H ⊂ RatCurves^{n}(X) is an irreducible component, it is known that H is proper if there exists a line bundle L ∈ Pic(X) that intersects a curve ℓ ∈ H with multiplicity L.ℓ = 1.

For complex projective manifolds we give another result. To formulate the setup properly, pick an irreducible component H ⊂ RatCurves^{n}(X) such that

(1) the rational curves associated with H dominate X,

(2) for a general point x ∈, the closed subfamily H_{x} is proper. Let Ũ be the universal family, which is a ℙ^{1}-bundle over H. The tangent map of the natural projection ι : Ũ → X, restricted to the relative tangent sheaf T_{Ũ/H}, gives rise to a rational map τ:

\xymatrix{ & ℙ(T_{X}^{∨}) \ar[d] \\ Ũ \ar@{-->}[ur]^{τ} \ar[r]^{ι}_{evaluation} \ar[d]_{π}^{ℙ^{1}-bundle} & X \\ H}

It has been shown in [Keb02a] that τ is generically finite. Examples of rationally connected manifolds, however, seem to suggest that the tangent map τ is generically injective for a large class of varieties. Our main result supports this claim.

Theorem 1.4. Let X be a smooth projective variety over the field of complex numbers and let H ⊂ RatCurves^{n}(X) be the union of irreducible components such that the subfamily H_{x} is proper for all points x ∈ X, outside a subvariety S ⊂ X of codimension at least 2.

Then τ is generically injective, unless the curves associated with the closed subfamily H^{cusp} ⊂ H of cuspidal curves dominate X, and the subvariety

D := {x ∈ X | ∃ℓ ∈ H^{cusp} : ℓ has a cuspidal singularity at x },

where curves have cuspidal singularities, has codimension 1.

Remark 1.4.1. It is known that the family H_{x} is proper for a general point x ∈ X if H is a “maximal dominating family of rational curves of minimal degrees”, i.e., if the degrees of the curves associated with H are minimal among all irreducible components of RatCurves^{n}(X) which satisfy condition (1) from above.

The assumption that H_{x} is proper for all points outside a set of codimension 2, however, is restrictive.

Although we consider these results only a first step toward a satisfactory answer to Question 1.2, we are already able to present several applications in Section 5.

Acknowledgements. Parts of this paper have been worked out while the first named author visited the University of Washington at Seattle, the University of British Columbia at Vancouver and Princeton University as well as while the second named author visited the Isaac Newton Institute for Mathematical Sciences at Cambridge. Both authors are grateful to these institutions for their hospitality. S. Kebekus would like to thank K. Behrend and J. Kollár for the invitations and for numerous discussions. S. Kovács would like to thank the organizers, Alessio Corti, Mark Gross, and Miles Reid for the invitation to the ’Higher Dimensional Geometry Programme’ of the Newton Institute.

After the main part of this paper was written, J.-M. Hwang has informed us that, together with N. Mok, they have shown a statement similar to, but somewhat stronger than Theorem 1.4. Their unpublished proof uses entirely different methods. To the best of our knowledge, there is no other result similar to Theorem 1.3.

2. ℙ^{1}-bundles with double sections

This preliminary section discusses ℙ^{1}-bundles with an irreducible double section. Most results here are fairly elementary. We have, however, chosen to include detailed proofs for lack of a suitable reference.

Throughout the present section let λ : Λ → B be a ℙ^{1}-bundle over a normal variety B, i.e., a morphism whose scheme-theoretic fibers are all isomorphic to ℙ^{1}. Note that we do not assume here that λ is trivial in the Zariski topology. Let σ : B → Λ be a section of λ, Σ_{red} = σ(B)_{red} ⊂ Λ, and let Σ ⊂ Λ be the first infinitesimal neighborhood of Σ_{red} in Λ. That is, if Σ_{red} is defined by the sheaf of ideals J = O_{Λ}(−Σ_{red}), then Σ is defined by the sheaf J^{2}. Our aim is to relate properties of Λ with those of its subscheme Σ.

2.1. The Picard group of the double section. Recall from [Har77, III. Ex.4.6] qthat there exists a short exact sequence of sheaves of Abelian groups, sometimes called that “truncated exponential sequence” in the literature (eg. [BBI00, sect. 2])

(2.1)  0 ⟶ \underbrace{J /J^{2}}_{=N_{Σ_{red}|Λ}^{∨}} ⟶^{α} O_{Σ}^{∗} ⟶^{β} O_{Σ_{red}}^{∗} ⟶ 1.

Here N_{Σ_{red}|Λ}^{∨} is the conormal bundle, β is the canonical restriction map and α is given by

\begin{aligned}α : & (J /J^{2}, +) & → & O_{Σ}^{∗} \\ & f & ↦ & 1 + f. \end{aligned}

In our setup, where Σ_{red} ≃ B is a section, the truncated exponential sequence (2.1) is canonically split. Locally we can write the splitting as follows. Assume that we are given an affine open subset U_{α} ⊂ Σ and an invertible function f_{α} ∈ O_{Σ}^{∗}(U_{α}). Then, after shrinking U_{α}, if needed, we will find a bundle coordinate y_{a}, identify

O_{Σ}^{∗}(U_{α}) ≃ [O_{Σ_{red}}(U_{α}) ⊗ k[y_{α}]/(y_{α}^{2})]^{∗}

and write accordingly

f_{α} = g_{α} + h_{α} · y_{α}

where g_{α} ∈ O_{Σ_{red}}^{∗}(U_{α}) and h_{α} ∈ O_{Σ_{red}}(U_{α}). With this notation, the splitting of sequence (2.1) decomposes f_{α} as

f_{α} = g_{α} · \underbrace{[1 + \frac{h_{α}}{g_{α}} · y_{α}]}_{∈Im(α_{Uα})}

As a direct corollary to the splitting of (2.1) we obtain a canonical decomposition of the Picard group

(2.2)  Pic(Σ) = Pic(Σ_{red}) × H^{1}(Σ_{red}, N_{Σ_{red}|Λ}^{∨}).

2.2. The cohomology class of a line bundle. Let L ∈ Pic(Λ) be a line bundle. Using the decomposition (2.2) from above, we can associate to L a class c(L) ∈ H^{1}(Σ_{red}, N_{Σ_{red}|Λ}^{∨}). As this class will be important soon, we will now find a Čech-cocycle in Z^{1}(U_{α}, N_{Σ_{red}|Λ}^{∨}) that represents c(L).

To this end, find a suitable open affine cover U_{α} of Σ such that L|_{U_{α}} is trivial for all α and where bundle coordinates y_{α} exist. Let f_{α} ∈ L(U_{α}) be a collection of nowhere vanishing sections which we write in local coordinates as f_{α} = g_{α} + h_{α} · y_{α}. Using the U_{α}-coordinates on the intersection U_{α} ∩ U_{β}, the transition functions for the line bundle are thus written as

\frac{f_{α}}{f_{β}} = \frac{g_{α} + h_{α} · y_{α}}{g_{β} + h_{β} · y_{α}} = \frac{g_{α}}{g_{β}} · [1 + (\frac{h_{α}}{g_{α}} − \frac{h_{β}}{g_{β}} ) y_{α}] ∈ O_{Σ}^{∗}(U_{αβ})

In other words, the class of c(L) ∈ H^{1}(Σ_{red}, N_{Σ_{red}|Λ}^{∨}) is represented by the Čech cocycle

(2.3)  (\frac{h_{α}}{g_{α}} − \frac{h_{β}}{g_{β}} ) y_{α} ∈ Z^{1}(U_{α}, N_{Σ_{red}|Λ}^{∨})

2.3. Vector bundle sequences associated to line bundles. Consider the ideal sheaf sequence for Σ_{red} ⊂ Σ.

0 ⟶ J /J^{2} ⟶ O_{Σ} ⟶ O_{Σ_{red}} ⟶ 0

Warning 2.1. It should be noted that Σ_{red} is not a Cartier-divisor in Σ since its ideal sheaf, J /J^{2} ≃ N_{Σ_{red}|Λ}^{∨} is not a locally free O_{Σ}-module. Furthermore, the restriction of the ideal sheaf of Σ_{red} in Λ to Σ, J ⊗ O_{Σ} ≃ J /J^{3}, is not isomorphic to the ideal sheaf of Σ_{red} in Σ, J /J^{2} ≄ J ⊗ O_{Σ}. In fact, J ⊗ O_{Σ} is not even a subsheaf of O_{Σ}.

Construction 2.2. Let L ∈ Pic(Σ) be a line bundle. By abuse of notation, identify Σ_{red} with B and consider L|_{Σ_{red}} a line bundle on B. Then twist the above sequence with the locally free O_{Σ}-module L ⊗ λ^{∗}(L^{∨}|_{Σ_{red}}), and obtain the following sequence of O_{Σ}- modules,

(2.4)  0 ⟶ N_{Σ_{red}|Λ}^{∨} ⟶ L ⊗ λ^{∗}(L^{∨}|_{Σ_{red}}) ⟶ O_{Σ_{red}} ⟶ 0.

Finally, consider the push-forward to B:

(2.5)  0 ⟶ N_{Σ_{red}|Λ}^{∨} ⟶ \underbrace{λ_{∗}(L) ⊗ L^{∨}|_{Σ_{red}}}_{=:E_{L}} ⟶^{A} O_{Σ_{red}} ⟶ 0.

We obtain a vector bundle E_{L} of rank two on B which is presented as an extension of two line bundles. The surjective map E_{L} → O_{B} induces a section σ_{L} : B → ℙ(E). We will use this notation later and also extend it to line bundles, L ∈ Pic(Λ), by E_{L} := E_{L|_{Σ}} and σ_{L} = σ_{L|_{Σ}}. Observe that (ℙ(E_{L}), σ_{L}) depends on L only up to a twist by a line bundle pulled back from B. I.e., for M ∈ Pic(B), E_{L⊗λ^{∗}M} ≃ E_{L} and σ_{L⊗λ^{∗}M} = σ_{L}.

Much of our further argumentation is based on the following observation.

Proposition 2.3. Let L ∈ Pic(Σ) be a line bundle and c(L) ∈ H^{1}(Σ_{red}, N_{Σ_{red}|Λ}^{∨}) the class defined above. Then c(L) coincides with the extension class

e(L) ∈ Ext^{1}(O_{Σ_{red}}, N_{Σ_{red}|Λ}^{∨}) = H^{1}(Σ_{red}, N_{Σ_{red}|Λ}^{∨})

of the vector bundle sequence (2.5). In particular, the map

e : (Pic(Σ), ⊗) →  (H^{1}(Σ_{red}, N_{Σ_{red}|Λ}^{∨}), +)
L  ↦ extension class of sequence (2.5)

is a homomorphism of groups.

Proof. The proof relies on an explicit calculation in Čech cohomology. We will choose a sufficiently fine cover U_{α} of Σ_{red} and produce a Čech cocycle in Z^{1}(U_{α}, N_{Σ_{red}|Λ}^{∨}) that represents the extension class e(L). It will turn out that this cocycle equals the one that we have calculated in (2.3) above for c(L).

We keep the notation from above and let f_{α} ∈ L(U_{α}) be a collection of nowherevanishing sections of L. Such sections can be naturally seen to give local splittings of the sequences (2.4) and (2.5). Explicitly, if we write f_{α} = g_{α} + h_{α} · y_{α}, then

\frac{f_{α}}{g_{α}} = 1 + \frac{h_{α}}{g_{α}} · y_{α} ∈ (L ⊗ L^{∨}|_{Σ_{red}})(U_{α})

are nowhere-vanishing sections of L ⊗ L^{∨}|_{Σ_{red}} and the splitting takes the form

s_{α} : O_{Σ_{red}}(U_{α}) → (L ⊗ L^{∨}|_{Σ_{red}})(U_{α})
1  ↦  1 + \frac{h_{α}}{g_{α}} · y_{α}

By construction of Ext^{1}, we obtain the extension class as the homology class represented by the Čech cocycle

s_{α}(1) − s_{β}(1) ∈ ker(A)(U_{αβ}) ≃ N_{Σ_{red}|Λ}^{∨}(U_{αβ})

This difference is given by the following section in N_{Σ_{red}|Λ}^{∨}(U_{αβ}) which yields the required cocycle.

(1 + \frac{h_{α}}{g_{α}} · y_{α}) − (1 + \frac{h_{β}}{g_{β}} · y_{α}) = (\frac{h_{α}}{g_{α}} − \frac{h_{β}}{g_{β}} ) y_{α} ∈ Z^{1}(U_{α}, N_{Σ_{red}|Λ}^{∨})

That, however, is the same cocycle which we have obtained above in formula (2.3) for the class c(L). The proof of Proposition 2.3 is therefore finished. □

2.4. The reconstruction of the ℙ^{1}-bundle from a double section. It is a remarkable fact that the restriction of an ample line bundle L ∈ Pic(Λ) to a double section carries enough information so that the whole ℙ^{1}-bundle Λ can be reconstructed. The proof is little more than a straightforward application of Proposition 2.3. We are grateful to Ivo Radloff who showed us how to use extension classes to simplify our original proof.

Notation 2.4. Let (Λ, σ) and (Λ′, σ′) be two ℙ^{1}-bundles with sections over B. We say that (Λ, σ) and (Λ′, σ′) are isomorphic pairs (over B) if there exists a morphism γ : Λ → Λ′, an isomorphism of pairs, such that γ is a B-isomorphism of ℙ^{1}-bundles and γ ∘ σ = σ′. Sometimes we will refer to these pairs by the image of the section: (Λ, σ(B)), in which case the meaning of isomorphic pairs should be clear.

Theorem 2.5. Given a line bundle L ∈ Pic(Λ), which is not the pull-back of a line bundle on B, let E_{L} and σ_{L} be as in 2.2. Consider the relative degree d ∈ ℤ\{0} of L, i.e., the intersection number with fibers of λ. If d is coprime to char(k), then (Λ, σ) and (ℙ(E_{L}), σ_{L}) are isomorphic pairs over B.

Proof. Let H := O_{Λ}(Σ_{red}) = J^{∨}. Then Λ ≃ ℙ(λ_{∗}H) and σ : B → Λ is the section associated to the surjection, λ_{∗}H → λ_{∗}(H|_{Σ_{red}}).

First we would like to prove that λ_{∗}H ≃ λ_{∗}(H|_{Σ}). Indeed, consider the sequence,

0 ⟶ H ⊗ J^{2} ≃ J ⟶ H ⟶ H|_{Σ} ⟶ 0.

We need to prove that λ_{∗}J ≃ R^{1}λ_{∗}J ≃ 0. However, that follows from considering the pushforward of the sequence,

0 ⟶ J ⟶ O_{Λ} ⟶ O_{Σ_{red}} ⟶ 0,

since λ_{∗}O_{Λ} ≃ λ_{∗}O_{Σ_{red}} ≃ O_{B}, and R^{1}λ_{∗}O_{Λ} ≃ 0.

This implies the statement for L = H, that is, we obtain that (Λ, σ) and (ℙ(E_{H}), σ_{H}) are isomorphic pairs over B (cf. [Har77, II.7.9]).

In order to finish the proof, we are going to prove that (ℙ(E_{H}), σ_{H}) and (ℙ(E_{L}), σ_{L}) are isomorphic pairs over B for any L ∈ Pic(Λ). In fact, it suffices to show that the extension classes of the following sequences are the same up to a non-zero scalar multiple.

(2.6)

0 ⟶ N_{Σ_{red}|Λ}^{∨} ⟶ λ_{∗}(H|_{Σ}) ⊗ H^{∨}|_{Σ_{red}} ⟶ O_{Σ_{red}} ⟶ 0
0 ⟶ N_{Σ_{red}|Λ}^{∨} ⟶ λ_{∗}(L|_{Σ}) ⊗ L^{∨}|_{Σ_{red}} ⟶ O_{Σ_{red}} ⟶ 0.

Recall that Pic(Λ) = ℤ × Pic(B) so that we can write L ∈ H^{⊗d} ⊗ λ^{∗}M for an appropriate M ∈ Pic(B). By Proposition 2.3 this implies that the extension classes of the sequences (2.6)are given by c (H|_{Σ}) and c (H|_{Σ}^{⊗d}) = d · c (H|_{Σ}). In particular, they differ only by the non-zero factor d ∈ k. □

Warning 2.6. The construction of the vector bundle E_{L} and Proposition 2.3 use only the restriction L|_{Σ}. It may thus appear that Theorem 2.5 could be true without the assumption that L ∈ Pic(Λ) and that one could allow arbitrary line bundles L ∈ Pic(Σ) instead. That, however, is wrong and counterexamples do exist. Note that the proof of Theorem 2.5 uses the fact that L is contained in ℤ × Pic(B) which is not true in general if L ∈ Pic(Σ) is arbitrary.

The assumption that d be coprime to char(k) is actually necessary in Theorem 2.5, as shown by the following simple corollary of Proposition 2.3 and of the proof of Theorem 2.5.

Corollary 2.7. Using the same notation as in Theorem 2.5, assume that d is divisible by char(k). Then

λ_{∗}(L|_{Σ}) ⊗ L^{∨}|_{Σ_{red}} ≃ N_{Σ_{red}|Λ}^{∨} ⊕ O_{Σ_{red}}.

3. Dubbies

Throughout the proofs of Theorems 1.3 and 1.4, which we give in Sections 4.1 and 4.2 below, we will assume that X contains pairs of minimal rational curves which intersect tangentially in at least one point. A detailed study of these pairs and their parameter spaces will be given in the present chapter. The simplest configuration is the following:

Definition 3.1. A dubby is a reduced, reducible curve, isomorphic to the union of a line and a smooth conic in ℙ^{2} intersecting tangentially in a single point.

Remark 3.1.1. The definition may suggest at first glance that one component of a dubby is special in that it has a higher degree than the other. We remark that this is not so. A dubby does not come with a natural polarization. In fact, there exists an involution in the automorphism group that swaps the irreducible components.

Later we will need the following estimate for the dimension of the space of global sections of a line bundle on a dubby. Let ℓ = ℓ_{1} ∪ ℓ_{2} be a dubby and L ∈ Pic(ℓ) a line bundle. We say that L has type (d_{1}, d_{2}) if the restrictions of L to the irreducible components ℓ_{1} and ℓ_{2} have degree d_{1} and d_{2}, respectively.

Lemma 3.2. Let ℓ be a dubby and L ∈ Pic(ℓ) a line bundle of type (d_{1}, d_{2}). Then h^{0}(ℓ, L) ≥ d_{1} + d_{2}.

Proof. By assumption, we have that L|_{ℓ_{i}} ≃ O_{ℙ^{1}}(d_{i}). Let ℓ_{1}.ℓ_{2} be the scheme theoretic intersection of ℓ_{1} and ℓ_{2}, ι^{i} : ℓ_{i} → ℓ the natural embedding, and L_{i} = ι_{∗}^{i}(L|_{ℓ_{i}}) for i = 1, 2. Then one has the following short exact sequence:

0 → L → L_{1} ⊕ L_{2} → O_{ℓ_{1}.ℓ_{2}} → 0.

This implies that h^{0}(ℓ, L) ≥ χ(L) = χ(L_{1}) + χ(L_{2}) − χ(O_{ℓ_{1}.ℓ_{2}}) = d_{1} + d_{2}. □

3.1. The identification of the components of a dubby. To illustrate the main observation about dubbies, let us consider a very simple setup first: let L ∈ Pic(X) be an ample line bundle, and assume that ℓ = ℓ_{1} ∪ ℓ_{2} ⊂ X is a dubby where both components are members of the same connected family H of minimal rational curves. In particular, L|_{ℓ} will be of type (d, d), where d > 0. Remarkably, the line bundle L induces a canonical identification of the two components ℓ_{1} and ℓ_{2}, at least when d is coprime to the characteristic of the base field k. Over the field of complex numbers, the idea of construction is the following: Fix a trivialization t : L|_{V} → O_{V} of L on an open neighborhood V of the intersection point {z} = ℓ_{1} ∩ ℓ_{2}. Given a point x ∈ ℓ_{1} \ ℓ_{2}, let σ_{1} ∈ H^{0}(ℓ_{1}, L|_{ℓ_{1}}) be a non-zero section that vanishes at x with multiplicity m. Then there exists a unique section σ_{2} ∈ H^{0}(ℓ_{2}, L|_{ℓ_{2}}) with the following properties:

(1) The section σ_{2} vanishes at exactly one point y ∈ ℓ_{2}.

(2) The sections σ_{1} and σ_{2} agree on the intersection of the components:

σ_{1}(z) = σ_{2}(z)

(3) The differentials of σ_{1} and σ_{2} agree at z:

v⃗(t ∘ σ_{1}) = v⃗(t ∘ σ_{2})

for all non-vanishing tangent vectors v⃗ ∈ T_{ℓ_{1}} ∩ T_{ℓ_{2}}. The map that associates x to y gives the identification of the components and does not depend on the choice of t.

In the following section 3.2, we will give a construction of the identification morphism which also works in the relative setup, for bundles of type (d_{1}, d_{2}) where d_{1} ≠ d_{2}, and in arbitrary characteristic.

3.2. Bundles of dubbies. For the proof of the main theorems we will need to consider bundles of dubbies, i.e., morphisms where each scheme-theoretic fiber is isomorphic to a dubby. The following Proposition shows how to identify the components of such bundles.

Proposition 3.3. Let λ : Λ → B be a projective family of dubbies over a normal base B and assume that Λ is not irreducible. Then it has exactly two irreducible components Λ_{1} and Λ_{2}, both ℙ^{1}-bundles over B. Assume further that there exists a line bundle L ∈ Pic(Λ) whose restriction to a λ-fiber has type (m, n′), where m and n are non-zero and relatively prime to char(k).

If Σ_{red} ⊂ Λ_{1} ∩ Λ_{2} denotes the reduced intersection, then Σ_{red} is a section over B, and the pairs (Λ_{1}, Σ_{red}) and (Λ_{2}, Σ_{red}) are isomorphic over B.

Note that the isomorphism given in Proposition 3.3 is not canonical and may not respect the line bundle L.

Proof of Proposition 3.3. The map λ is flat because all its scheme-theoretic fibers are isomorphic. Let Λ_{1} ⊂ Λ be one of the irreducible components. It is easy to see that if x ∈ Λ_{1} is a general point, then Λ_{1} contains the (unique) irreducible component of ℓ_{λ(x)} := λ^{−1}λ(x) that contains x. Since λ is proper and flat, λ(Λ_{1}) = B. Hence Λ_{1} contains one of the irreducible components of ℓ_{b} for all b ∈ B. Repeating the same argument with another irreducible component, Λ_{2}, one finds that it also contains one of the irreducible components of ℓ_{b} for all b ∈ B. However, they cannot contain the same irreducible component for any b ∈ B: In fact, if they contained the same component of ℓ_{b} for infinitely many points b ∈ B, then they would agree. On the other hand, if they contained the same component of ℓ_{b} for finitely many points b ∈ B, then Λ would have an irreducible component that does not dominate B. This, however, would contradict the flatness of λ. Hence Λ_{1} ∪ Λ_{2} = Λ. They are both ℙ^{1}-bundles over B by [Kol96, Thm. II.2.8.1].

Let Σ := Λ_{1} ∩ Λ_{2} be the scheme-theoretic intersection. Since Λ is a bundle of dubbies and B is normal, it is clear that its reduction, Σ_{red} is a section, and that Σ is its first infinitesimal neighborhood in either Λ_{1} or Λ_{2}. In this setup, the isomorphism of pairs is given by Theorem 2.5. □

3.3. The space of dubbies. In addition to the space of rational curves, which we use throughout, it is also useful to have a parameter space for dubbies. For the convenience of the reader, we will first recall the construction of the former space very briefly. The reader is referred to [Kol96, chapt. II.1] for a thorough treatment.

3.3.1. The space of rational curves. Recall that there exists a scheme Hom_{bir}(ℙ^{1}, X) whose geometric points correspond to morphisms ℙ^{1} → X that are birational onto their images. Furthermore, there exists an “evaluation morphism”: μ : Hom_{bir}(ℙ^{1}, X) × ℙ^{1} → X. The group PGL_{2} acts on the normalization Hom_{bir}^{n}(ℙ^{1}, X), and the geometric quotient exists. More precisely, we have a commutative diagram

\xymatrix{Hom_{bir}^{n}(ℙ^{1}, X) × ℙ^{1} \ar[d] \ar[r]^{U} \ar[rr]^{μ} & Univ^{rc}(X) \ar[r]^{ι} \ar[d]_{π} & X \\ Hom_{bir}^{n}(ℙ^{1}, X) \ar[r]^{u} & RatCurves^{n}(X)}

where u and U are principal PGL_{2} bundles, π is a ℙ^{1}-bundle and the restriction of the “evaluation morphism” ι to any fiber of π is a morphism which is birational onto its image. The quotient space RatCurves^{n}(X) is then the parameter space of rational curves on X. The letter “n” in RatCurves^{n} may be a little confusing. It has nothing to do with the dimension of X and it’s not a power. It serves as a reminder that the parameter space is the normalization of a suitable quasiprojective subset of the Chow variety.

It may perhaps look tempting to define a space of dubbies in a similar manner, as a quotient of the associated Hom-scheme. However, since geometric invariant theory becomes somewhat awkward for group actions on non-normal varieties, we have chosen another, elementary but somewhat lengthier approach. The space of dubbies will be constructed as a quasi-projective subvariety of the space of ordered pairs of pointed rational curves, and the universal family of dubbies will be constructed directly.

3.3.2. Pointed rational curves. It is easy to see that RC_{•}(X) = Univ^{rc}(X) naturally parametrizes pointed rational curves on X and the pull-back of the universal family

Univ_{•}^{rc}(X) = RC_{•}(X) ×_{RatCurves^{n}(X)} Univ^{rc}(X)

is the universal family of pointed rational curves over RC_{•}(X). The identification morphism RC_{•}(X) → Univ^{rc}(X) and the identity map of RC_{•}(X) gives a section of this universal family:

\xymatrix{Univ_{•}^{rc}(X) \ar[d] \ar[r] & Univ^{rc}(X) \ar[d] \\ RC_{•}(X) \ar@{.>}[u]^{η} \ar[ur]_{≃} \ar[r]_{π} & RatCurves^{n}(X)}

3.3.3. Pairs of pointed rational curves. The product RC_{•}^{2}(X) := RC_{•}(X) × RC_{•}(X) naturally parametrizes pairs of pointed rational curves. We denote the projections to the two factors by ρ_{i} : RC_{•}^{2}(X) → RC_{•}(X) for i = 1, 2. Then the universal family will be given as the disjoint union

Univ_{•}^{rc,2}(X) = (RC_{•}^{2}(X) ×_{ρ_{1}} Univ_{•}^{rc}(X)) ∪ (RC_{•}^{2}(X) ×_{ρ_{2}} Univ_{•}^{rc}(X)) .

The two copies of the section η : RC_{•}(X) → Univ_{•}^{rc}(X) induce two sections of this family, one for each component of the union:

\xymatrix{ & Univ_{•}^{rc,2}(X) \ar[dl] \ar[dd]_{p̃} \ar[dr] \\ Univ_{•}^{rc}(X) \ar[dd] &  & Univ_{•}^{rc}(X) \ar[dd] \\  & RC_{•}^{2}(X) \ar[dl]_{ρ_{1}} \ar[dr]^{ρ_{2}} \ar@{.>}[uu]^{σ_{1}} \ar@{.>}[uu]_{σ_{2}} \\ RC_{•}(X) \ar@{.>}[uu]^{η} &  & RC_{•}(X) \ar@{.>}[uu]_{η}}

3.3.4. The space of dubbies. Consider the evaluation morphism ι^{2} : Univ_{•}^{rc,2}(X) → X. The associated tangent map T ι^{2} restricted to the relative tangent sheaf T_{Univ_{•}^{rc,2}(X)/ RC_{•}^{2}(X)} gives rise to a rational map

τ^{rc,2} : Univ_{•}^{rc,2}(X) −−→ ℙ(T_{X}^{∨}).

We define a quasiprojective variety, the space of dubbies,

Dubbies^{n}(X) := normalization of {ℓ ∈ RC_{•}^{2}(X) | τ^{rc,2} is defined at σ_{1}(ℓ)

and at σ_{2}(ℓ), and τ^{rc,2}(σ_{1}(ℓ)) = τ^{rc,2}(σ_{2}(ℓ))}.

We will often consider pairs of curves such that both components come from the same family H ⊂ RatCurves^{n}(X). For this reason we define π_{2} : Dubbies^{n}(X) → RatCurves^{n}(X) × RatCurves^{n}(X), the natural forgetful projection morphism, and

Dubbies^{n}(X)|_{H} := Dubbies^{n}(X) ∩ π_{2}^{−1}(H × H).

Proposition 3.4. Assume that H ⊂ RatCurves^{n}(X) is a proper family of immersed curves. Then Dubbies^{n}(X)|_{H} is also proper.

Proof. Since the tangent map, τ^{rc,2}, is well-defined at σ_{1}(ℓ) and σ_{2}(ℓ) for every ℓ ∈ RC_{•}^{2}(X) ∩ π_{2}^{−1}(H × H),

Dubbies^{n}(X) = normalization of {ℓ ∈ RC_{•}^{2}(X) | τ^{rc,2}(σ_{1}(ℓ)) = τ^{rc,2}(σ_{2}(ℓ))},

which is clearly a closed subvariety of the proper variety π_{2}^{−1}(H × H). □

The next statement follows immediately from the construction and from the universal property of RatCurves^{n}(X).

Proposition 3.5. Let ℓ_{1} and ℓ_{2} ⊂ X be rational curves with normalizations

η_{i} : ℙ^{1} ≃ ℓ̃_{i} → ℓ_{i} ⊂ X.

If T η_{i} have rank 1 at the point [0 : 1] ∈ ℙ^{1} for i = 1, 2, and if the images of the tangent morphisms agree,

Image(T η_{1}|_{[0:1]}) = Image(T η_{2}|_{[0:1]}) ⊂ T_{X},

then there exists a point ℓ ∈ Dubbies^{n}(X) such that p̃^{−1}(ℓ) = ℓ̃_{1} ∪ ℓ̃_{2}.

α [figure]

ι

[figure]

two disjoint ; ℙ^{1}-bundles

p̃

[figure]

β [figure]

p̂

bundle of dubbies

[figure]

p bundle of two rational curves with complicated intersection

[figure]

Figure 3.1. A partial resolution of singularities

If H ⊂ RatCurves^{n}(X) is a subfamily, and both ℓ_{i} correspond to points of H, then we can find such an ℓ in Dubbies^{n}(X)|_{H}. □

3.3.5. The universal family of dubbies. In order to show that Dubbies^{n}(X) is a space of dubbies indeed, we need to construct a univeral family, which is a bundle of dubbies in the sense of section 3.2. To this end, we will factor the universal evaluation morphism.

Proposition 3.6. The evaluation morphism,

ι : \underbrace{Univ_{•}^{rc,2}(X) ×_{RC_{•}^{2}(X)} Dubbies^{n}(X)}_{=:Ũ, decomposes as Ũ_{1}∪Ũ_{2}} ⟶ U ⊂ X × Dubbies^{n}(X),

factors as follows:

(3.1)  \xymatrix{Ũ \ar[r]^{α} \ar[rr]^{ι} \ar[dr]_{two disjoint ℙ^{1}-bundles Ũ_{1} ∪ Ũ_{2}}^{p̃} & Λ \ar[r]^{β} \ar[d]^{p̂}_{bundle of dubbies} & U \ar[dl]_{p}^{bundle of two curves with complicated intersection} \\  & Dubbies^{n}(X)}

where the variety Λ is a bundle of dubbies in the sense that for every closed point b ∈ B^{0}, the fiber p̂^{−1}(b) is isomorphic to a dubby.

Remark 3.6.1. If ℓ ∈ Dubbies^{n}(X) is any point, then the two corresponding curves in X intersect tangentially in one point, but may have very complicated intersection at that point and elsewhere. The factorization of the evaluation morphism should therefore be understood as a partial resolution of singularities, as shown in figure 3.1.

Proof. As a first step we will construct the space Λ. Because the evaluation ι is a finite, hence affine, morphism, it seems appropriate to construct a suitable subsheaf A ⊂ ι_{∗}OŨ, which is a coherent sheaf of O_{U}-modules and set β : Λ = Spec(A) → U.

Let σ̄_{1} ⊂ Ũ_{1} and σ̄_{2} ⊂ Ũ_{2} be the images of the pullbacks of the canonical sections, σ_{1} and σ_{2}, constructed in 3.3.3. In order to construct A, we will need to find an identification of their first infinitesimal neighborhoods, σ̃_{1} and σ̃_{2}. Since ι is separable, it follows directly from the construction that σ̃_{1} and σ̃_{2} map isomorphically onto their scheme-theoretic images ι(σ̃_{1}) and ι(σ̃_{2}). Again, by the definition of Dubbies^{n}(X), these images agree: ι(σ̃_{1}) = ι(σ̃_{2}) and we obtain the desired identification,

γ : σ̃_{1} → σ̃_{2}.

Let

i_{1} : σ̃_{1} → Ũ_{1} and i_{2} : σ̃_{2} → Ũ_{2}

be the inclusion maps and consider the sheaf morphism

φ := ι_{∗}(i_{1}^{#} − γ^{#} ∘ i_{2}^{#}) : ι_{∗}OŨ → ι_{∗}Oσ̃_{1}.

The sheaf

A := ker(φ)

is thus a coherent sheaf of O_{U}-modules. As it was planned above, define Λ := Spec(A). The existence of the morphisms α and β and that ι = β ∘ α follows from the construction. It remains to show that Λ is a bundle of dubbies. Let ℓ ∈ Dubbies^{n}(X) be a closed point. Replacing Dubbies^{n}(X) with a neighbourhood of ℓ and passing to a finite, unbranched cover if necessary, and by abuse of notation still denoting it by Dubbies^{n}(X), we can assume that

(1) the variety Dubbies^{n}(X) is affine, say Dubbies^{n}(X) ≃ Spec R,

(2) the ℙ^{1}-bundles Ũ_{i} = ℙ(p̃_{∗}OŨ_{i}(σ̄_{i})^{∨}), for i = 1, 2 are trivial, and

(3) there exists a Cartier divisor τ ⊂ U such that ι^{−1}(τ) = τ_{1} ∪ τ_{2}, where τ_{i} ⊂ Ũ_{i}

are sections that are disjoint from σ̄_{i}. We can then find homogeneous bundle coordinates [x_{0} : x_{1}] on Ũ_{1} and [y_{0} : y_{1}] on Ũ_{2} such that

σ̄_{1} = {([x_{0} : x_{1}], b) ∈ Ũ_{1} | x_{0} = 0},  τ_{1} = {([x_{0} : x_{1}], b) ∈ Ũ_{1} | x_{1} = 0},
σ̄_{2} = {([y_{0} : y_{1}], b) ∈ Ũ_{2} | y_{0} = 0}, and τ_{2} = {([y_{0} : y_{1}], b) ∈ Ũ_{2} | y_{1} = 0}.

If we set

Ũ_{0} := Ũ \ (τ_{1} ∪ τ_{2}),

then the image U_{0} := ι(Ũ_{0}) is affine, and we can write the relevant modules as

OŨ(Ũ_{0}) ≃ R ⊗ (k[x_{0}] ⊕ k[y_{0}])
Oσ̃_{1}(Ũ_{0}) ≃ R ⊗ k[x_{0}]/(x_{0}^{2}).
Oσ̃_{2}(Ũ_{0}) ≃ R ⊗ k[y_{0}]/(y_{0}^{2}).

Adjusting the bundle coordinates, if necessary, we can assume that the identification morphism γ^{#}(U_{0}) : Oσ̃_{2}(Ũ_{0}) → Oσ̃_{1}(Ũ_{0}) is written as

\begin{aligned}γ^{#}(U_{0}) : & R ⊗ k[y_{0}]/(y_{0}^{2}) & → & R ⊗ k[x_{0}]/(x_{0}^{2}) \\ & r ⊗ y_{0} & ↦ & r ⊗ x_{0}. \end{aligned}

In this setup, we can find the morphism φ explicitly:

\begin{aligned}φ(U_{0}) : & R ⊗ (k[x_{0}] ⊕ k[y_{0}]) & → & R ⊗ k[x_{0}]/(x_{0}^{2}) \\ & r ⊗ (f(x_{0}), g(y_{0})) & ↦ & r ⊗ (f(x_{0}) − g(x_{0})). \end{aligned}

[figure: {Λ_{T} \atop τ_{1}}; pr_{1}∘β; p̂; X x •; T]

Figure 4.1. Proof of Theorem 1.4

Therefore, as an R-algebra, ker(φ)(U_{0}) is generated by the elements u := 1_{R} ⊗ (x_{0}, y_{0}) and v := 1_{R} ⊗ (x_{0}^{2}, 0), which satisfy the single relation v(u^{2} − v) = 0. Thus

ker(φ)(U_{0}) = R ⊗ k[u, v]/(v(u^{2} − v)).

In other words, β^{−1}(U_{0}) is a bundle of two affine lines over Dubbies^{n}(X), meeting tangentially in a single point.

It follows directly from the construction of A that α is an isomorphism away from σ̄_{1} ∪ σ̄_{2}. The curve p̂^{−1}(ℓ) is therefore smooth outside of p̂^{−1}(ℓ) ∩ β^{−1}(U_{0}), and it follows that p̂^{−1}(ℓ) is indeed a dubby. This ends the proof. □

4. Proofs of the Main Theorems

4.1. Proof of Theorem 1.3. The assertion that all curves associated with H_{x} are smooth at a general point x ∈ X follows immediately from the assumption that none of the curves ℓ ∈ H is cuspidal, and by [Keb02b, thms. 2.4(1) and 3.3(1)]. It remains to show that no two curves intersect tangentially.

We will argue by contradiction and assume that we can find a pair ℓ = ℓ_{1} ∪ ℓ_{2} ⊂ X of distinct curves ℓ_{i} ∈ H that intersect tangentially at x. The pair ℓ is then dominated by a dubby whose singular point maps to x. Loosely speaking, we will move the point of intersection to obtain a positive-dimensional family of dubbies that all contain the point x –see figure 4.1.

Setup. To formulate our setup more precisely, we will use the notation introduced in diagram (3.1) of Proposition 3.6 and recall from Proposition 3.4 that Dubbies^{n}(X)|_{H} is proper. Recall further that the universal family U is a subset U ⊂ X × Dubbies^{n}(X) and let ȷ := pr_{1}∘ι : Ũ → X be the canonical morphism. The assumption that for every general point x ∈ X, there is a pair of curves intersecting tangentially at x can be reformulated as

ȷ ∘ σ_{1}(Dubbies^{n}(X)|_{H}) = ȷ ∘ σ_{2}(Dubbies^{n}(X)|_{H}) = X

Let D ⊂ Dubbies^{n}(X)|_{H} be an irreducible component such that

ȷ ∘ σ_{1}(D) = ȷ ∘ σ_{2}(D) = X

holds. By abuse of notation, we will denote Ũ_{D} = (Ũ_{D})_{1} ∪ (Ũ_{2})_{D} by Ũ = Ũ_{1} ∪ Ũ_{2}. Fix a closed point t ∈ D and consider the intersection numbers

d_{1} := ȷ^{∗}(L) · (p̃^{−1}(t) ∩ Ũ_{1}) and d_{2} := ȷ^{∗}(L) · (p̃^{−1}(t) ∩ Ũ_{2})

Renumbering Ũ_{1} and Ũ_{2}, if necessary, we may assume without loss of generality that d_{1} ≥ d_{2}. In this setup it follows from the upper semi-continuity of the fiber dimension that (ȷ|Ũ_{1})^{−1}(x) contains an irreducible curve τ_{1} which intersects σ_{1}(D) non-trivially and is not contained in

S := {y ∈ Ũ | ι is not an isomorphism at y}

Set T := p̃(τ_{1}). After a base change, if necessary, we may assume that T is a normal curve and consider the restrictions of the morphisms constructed in Proposition 3.6:

\xymatrix{Ũ_{T} \ar[r]_{α} \ar[rrr]^{ȷ} & Λ_{T} \ar[r]_{β} & U_{T} \ar[r]_{pr_{1}} & X}

Using [Keb02a, thm. 3.3.(1)], we find that τ_{1} is generically injective over T, and therefore is a section. Let Ũ_{T,1} = (Ũ_{1})_{T} and Ũ_{T,2} = (Ũ_{2})_{T}. By Proposition 3.3, (Ũ_{T,1}, σ_{1}(T)) and (Ũ_{T,2}, σ_{2}(T)) are isomorphic pairs over T. Let γ : Ũ_{T,1} → Ũ_{T,2} be an isomorphism and consider the section τ_{2} := γ(τ_{1}) ⊂ Ũ_{T,2}.

The contraction of τ_{2}. With the notation above, Theorem 1.3 follows almost immediately from the following observation.

Lemma 4.1. The morphism ȷ contracts the section τ_{2} to x, i.e., τ_{2} ⊂ ȷ^{−1}(x).

Notice that this finishes the proof of Theorem 1.3. Indeed, Lemma 4.1 implies that a general point t ∈ T corresponds to a pair ℓ_{t} = ℓ_{t,1}∪ℓ_{t,2} of two distinct curves that intersect at x. The curve ℓ_{t} is then singular at x, a contradiction to the fact that τ_{1} ⊄ S.

Proof Lemma 4.1. As a first step, we show that ȷ contracts τ_{2} to some point y ∈ X. The proof relies on a calculation of intersection numbers on the ruled surfaces Ũ_{T,1} and Ũ_{T,2}. Recall the basic fact that

Num(Ũ_{T,1}) = ℤ · σ_{1}(T) ⊕ ℤ · F_{V,1}

where F_{V,1} is a fiber of p̃Ũ_{T,1} : Ũ_{T,1} → T. A similar decomposition holds for Ũ_{T,2}. Since τ_{1} is a section, we have the numerical equivalence,

τ_{1} ≡ σ_{1}(T) + d · F_{V,1},

where d is a suitable integer. Since γ maps σ_{1}(T) isomorphically onto σ_{2}(T), we obtain a similar equation on Ũ_{T,2},

τ_{2} ≡ σ_{2}(T) + d · F_{V,2}.

Next take the ample line bundle L ∈ Pic(X) and set

d_{3} := ȷ^{∗}(L) · σ_{1}(T) = ȷ^{∗}(L) · σ_{2}(T).

These two numbers are indeed equal since the evaluation morphism identifies the images of the two sections σ_{1}(T) and σ_{2}(T). Now we can write the intersection numbers as

ȷ^{∗}(L) · τ_{2} = ȷ^{∗}(L) · (σ_{2}(T) + d · F_{V,2})

= d_{3} + d · d_{2} = \underbrace{(d_{3} + d · d_{1})}_{=ȷ^{∗}(L)·τ_{1}=0} +d · (d_{2} − d_{1})

= d · (d_{2} − d_{1}) ≤ 0 Since L is ample, this shows that ȷ(τ_{2}) is a point, y ∈ X.

It remains to prove that x = y. In order to see that, it suffices to recall two facts. First, as it was already used above, the evaluation morphism identifies the images of the two sections σ_{1}(T) and σ_{2}(T). Second, we know that τ_{1} and the canonical section σ_{1} ⊂ Λ_{1} intersect. Let t ∈ p̃(τ_{1} ∩ σ_{1}(T)) be a closed point. The two sections τ_{2} and σ_{2}(T) will then also intersect, t ∈ p̃(τ_{2} ∩ σ_{2}(T)) and we obtain

x = ȷ(τ_{1}) = ȷ(τ_{1} ∩ σ_{1}(T) ∩ p̃^{−1}(t))
= ȷ(τ_{2} ∩ σ_{2}(T) ∩ p̃^{−1}(t)) = y.

Lemma 4.1 is thus shown. □

4.2. Proof of Theorem 1.4. Let H ⊂ RatCurves^{n}(X) be as in Theorem 1.4. We assume without loss of generality that all irreducible components of H dominate X. Fix an ample line bundle L ∈ Pic(X) and let H′ ⊂ H be an irreducible component such that for a general curve C ∈ H′ the intersection number L.C is minimal among all the intersection numbers of L with curves from H. Finally, fix a rational curve C ⊂ X that corresponds to a general point of H′.

The proof of Theorem 1.4 now follows very much the lines of the proof of Theorem 1.3 from the previous section. The main difference to the previous argument is that we have to work harder to find the family T, as the properness of Dubbies^{n}(X)|_{H} is no longer automatically guaranteed. Over the complex number field, however, the following lemma holds, which replaces the properness assumption in our context.

Lemma 4.2. Assume that X is a complex-projective manifold, and let S′ ⊂ X be a subvariety of codimension codim_{X} S′ ≥ 2. If C ∈ H is a curve that corresponds to a general point of H′, then C and S′ are disjoint: C ∩ S′ = ∅.

Proof. [Kol96, Chapt. II, Prop. 3.7 and Thm. 3.11] □

Corollary 4.3. Under the assumptions of Theorem 1.4, if C ∈ H′ is a general curve, and if codim_{X} D ≥ 2, then

H_{C} := {C′ ∈ H | C ∩ C′ ≠ ∅} ⊂ H

is proper, and the associated curves are immersed. In particular, C is immersed.

Proof. It suffices to note that C is disjoint from both S and D. □

Before coming to the proof of Theorem 1.4, we give a last preparatory lemma concerning the dimension of the locus D of cusps.

Lemma 4.4. If D ⊂ X is a divisor, then the subfamily H^{cusp} ⊂ H of cuspidal curves dominates X.

Proof. Argue by contradiction and assume that all cuspidal curves in H^{cusp} are contained in a divisor. The total space of the family of cuspidal curves is then at least (dim D + 1)- dimensional, so for a general point x ∈ D there exists a positive dimensional family of cuspidal curves that contain x and are contained in D. That, however, is impossible: it has been shown in [Keb02a, Thm. 3.3] that in the projective variety D, a general point is contained in no more then finitely many cuspidal curves. □

[figure: Λ_{T} ; Ũ_{T,2}; pr_{1}∘β; p̂; ℓ_{1} x •  C; T]

Figure 4.2. Proof of Theorem 1.4

Setup of the proof. For the proof of Theorem 1.4, we will again argue by contradiction. By Lemma 4.4 this amounts to the assumtion that τ is not generically injective, and that codim_{X} D ≥ 2. By Corollary 4.3, this implies that the space of curves which intersect C is proper and all associated curves are immersed along C. Since C was a general curve, the assumptions also imply that for a general point x ∈ C, there exists a point t ∈ Dubbies^{n}(X) corresponding to a pair of marked curves ℓ = ℓ_{1} ∩ ℓ_{2} such that ℓ_{2} = C and ℓ_{1} intersects C tangentially at x, i.e., Image(τ(σ_{1}(t))) = ℙ(T_{C}|_{x}^{∨}) where τ : Ũ → ℙ(T_{X}^{∨}) is the tangent morphism from the introduction. Hence we can find a proper curve T ⊂ Dubbies^{n}(X) with associated diagram

\xymatrix{Ũ_{T} \ar[r]_{α} \ar[rrr]^{ȷ} \ar[dr]_{p̃} & Λ_{T} \ar[r]_{β} \ar[d]^{p̂} & U_{T} \ar[r]_{pr_{1}} \ar[dl]^{p} & X \\  & T}

such that Ũ_{T} decomposes as Ũ_{T} = Ũ_{T,1} ∪ Ũ_{T,2}, where

Ũ_{T,2} ≃ C̃× T ≃ ℙ^{1} × T,

and where τ|_{σ_{1}(T)} dominates ℙ(T_{C}^{∨}).

End of proof. We are now in a situation which is very similar to the one considered in the proof of Theorem 1.3: we will derive a contradiction by calculating certain intersection numbers on Ũ_{T,1} and Ũ_{T,2}.

As a first step, remark that Ũ_{T,1} maps to a surface in X. It follows that ȷ^{∗}(L) is nef and big on Ũ_{T,1}.

Secondly, since Ũ_{T,2} is isomorphic to the trivial bundle ℙ^{1}×T, we have a decomposition

Num(Ũ_{T,2}) ≃ ℤ · F_{H,2} ⊕ ℤ · F_{V,2}

where F_{H,2} is the numerical class of a fiber of the map Ũ_{T,2} → ℙ^{1} and F_{V,2} that of a fiber of the map Ũ_{T,2} → T. Likewise, since the pairs (Ũ_{T,1}, σ_{1}(T)) and (Ũ_{T,2}, σ_{2}(T)) are isomorphic, let

Num(Ũ_{T,1}) ≃ ℤ · F_{H,1} ⊕ ℤ · F_{V,1}

be the corresponding decomposition. If d denotes the degree of the (finite, surjective) morphism

ȷ ∘ σ_{1} = ȷ ∘ σ_{2} : T → C,

then it follows directly from the construction that the curves of type F_{H,2} intersect σ_{2}(T) with multiplicity d. We obtain that

σ_{2}(T) ≡ F_{H,2} + d · F_{V,2} and thus σ_{1}(T) ≡ F_{H,1} + d · F_{V,1}.

To end the argumentation, let

d_{1} := ȷ^{∗}(L) · F_{V,1} and d_{2} := ȷ^{∗}(L) · F_{V,2}

In particular, we have that ȷ^{∗}(L) · σ_{2} = d · k_{2}. Recall that H′ ⊂ H was chosen so that d_{1} ≥ d_{2} and write:

ȷ^{∗}(L) · F_{H,1} = ȷ^{∗}(L) · (σ_{1}(T) − d · F_{V,1})
= d · d_{2} − d · d_{1}
≤ 0.

Because Ũ_{T,1} is covered by curves which are numerically equivalent to F_{H,1} that contradicts the assumption that ȷ^{∗}(L)|Ũ_{T,1} is big and nef. The proof of Theorem 1.4 is thus finished. □

5. Applications

5.1. Irreducibility Questions. Let H ⊂ RatCurves^{n}(X) be a maximal dominating family of rational curves of minimal degrees on a projective variety X. How many components can H have? If we pick an irreducible component H′ ⊂ H and fix a general point x ∈ X, does it follow that

H′_{x} := {ℓ ∈ H′ | x ∈ ℓ}

is irreducible? These questions have haunted the field for quite a while now, as the possibility that H′_{x} might be reducible poses major problems in many of the proposed applications of rational curves to complex geometry —see the discussion in [Hwa01].

It is conjectured [Hwa01, chap. 5, question 2] that the answers to both of these questions are affirmative for a large class of varieties. There exists particularly strong evidence if X is a complex manifold and if the dimension of H′_{x} is not too small. Theorem 1.3 enables us to give a partial answer.

Theorem 5.1. Under the assumptions of Theorem 1.3, if X is a complex manifold and if for a general point x ∈ X, and for all irreducible components H′ ⊂ H

dim H′_{x} ≥ \frac{dim X − 1}{2},

then H_{x} is irreducible. In particular, H is irreducible.

The main technical difficulty in proving Theorem 5.1 lies in the fact that the closed points of H are generally not in 1:1-correspondence with actual rational curves, a possibility that is sometimes overlooked in the literature. As a matter of fact, this correspondence is only generically injective, and it may well happen that two or more points of H correspond to the same curve ℓ ⊂ X. This is due to the very construction of the space RatCurves^{n}(X): recall from section 3.3 that RatCurves^{n}(X) is constructed as the quotient of the normalization of Hom_{bir}(ℙ^{1}, X). While Hom_{bir}(ℙ^{1}, X) is in 1:1correspondence with morphisms, ℙ^{1} → X, that are birational onto their imnage, the normalization morphism

Hom_{bir}^{n}(ℙ^{1}, X) → Hom_{bir}(ℙ^{1}, X),

need not be injective. For complex manifolds, however, we have the following workaround.

Lemma 5.2. Under the assumptions of Theorem 5.1, let x ∈ X be a general point and set

H_{x} := {ℓ ∈ H | x ∈ ℓ}

Then the closed points of H_{x} are in 1:1-correspondence with the associated curves in X.

Proof. Since x is a general point and since we have picked a fixed component, H′, all rational curves through x are free by the proof of [KMM92, thm. 1.1]. The space Hom_{bir}(ℙ^{1}, X) is therefore smooth at every point f ∈ Hom_{bir}(ℙ^{1}, X) whose image contains the point x by [Kol96, thm. II.1.7]. The normalization morphism

Hom_{bir}^{n}(ℙ^{1}, X) → Hom_{bir}(ℙ^{1}, X),

is thus isomorphic in a neighborhood of f. Since Hom_{bir}(ℙ^{1}, X) is in 1:1 correspondence with morphisms ℙ^{1} → X, the claim follows. □

This enables us to prove Theorem 5.1.

Proof of Theorem 5.1. Choose a general point x ∈ X, and let τ : H −−→ ℙ(T_{X}^{∨}) be the tangent morphism described in the introduction. Since all curves associated with H_{x} are smooth, τ restricts to a regular morphism

τ_{x} : H_{x} → ℙ(T_{X}^{∨}|_{x}).

This morphism is known to be finite [Keb02a, thm. 3.4]. By Theorem 1.3, τ_{x} is injective.

Now assume that H_{x} is not irreducible, H_{x} = H_{x,1} ∪ … ∪ H_{x,n}. Since τ_{x} is finite, we have that

dim(τ_{x}(H_{x,1})) + dim(τ_{x}(H_{x,2})) ≥ dim X − 1 = dim ℙ(T_{X}^{∨}|_{x})

Thus, by [Har77, thm. I.7.2],

τ_{x}(H_{x,1}) ∩ τ_{x}(H_{x,2}) ≠ ∅.

It follows that τ is not injective, a contradiction. □

Lemma 5.2 raises the following interesting question.

Question 5.3. Are there other conditions than smoothness over ℂ which guarantee that closed points of H_{x} are in 1:1-correspondence with rational curves?

5.2. Stability of the tangent bundle, Automorphism Groups. The setup of Theorem 5.1 naturally generalizes the notion of a prime Fano manifold, i.e., one that is covered by lines under a suitable embedding. Virtually all results that have been obtained for prime Fanos hold in the more general setup of Theorem 5.1. We give two examples.

Theorem 5.4. In the setup of Theorem 5.1, the tangent bundle T_{X} is stable. In particular, T_{X} is simple, i.e., the only endomorphisms of T_{X} are the scalar multiplications.

Proof. The proof of [Hwa01, thm. 2.11] applies verbatim. □

For any complex variety X, let Aut_{0}(X) denote the maximal connected subgroup of the group of automorphisms. By universal properties, an automorphism of a complex variety induces an automorphism of the space RatCurves^{n}(X). It might be interesting to note that in our setup the converse also holds.

Theorem 5.5. In the setup of Theorem 5.1, the groups Aut_{0}(X) and Aut_{0}(H) coincide.

Proof. The proof of [HM02, cor. 1] applies verbatim. □

5.3. Contact Manifolds. Let X be a projective contact manifold over ℂ, e.g. the twistor space over a Riemannian manifold with Quaternionic-Kählerian holonomy group and positive curvature. We refer to [Keb01c] and the references therein for an introduction and for the relevant background information.

If X is different from the projective space, it has been shown in [Keb01c] that X is covered by a compact family of rational curves H ⊂ RatCurves^{n}(X) such that for a general point x, all curves associated with points in H_{x} are smooth. Thus, the assumptions of Theorem 1.4 are satisfied, and τ is generically injective. This has been shown previously in [Keb01b] using rather involved arguments which heavily rely on obstructions to deformations coming from contact geometry.

References

[BBI00] L. Bădescu, M.C. Beltrametti and P. Ionescu. Almost-lines and quasi-lines on projective manifolds. In T. Peternell and F.O. Schreyer, editors, Complex Analysis and Algebraic Geometry, pages 1–27. de Gruyter, 2000.

[CMSB00] K. Cho, Y. Miyaoka, and N.I. Shepherd-Barron. Characterizations of Projective Spaces and Applications. Preprint, October–December 2000.

[Eis95] D. Eisenbud. Commutative Algebra with a View Toward Algebraic Geometry, volume 150 of Graduate Texts in Mathematics. Springer, 1995.

[Har77] R. Hartshorne. Algebraic Geometry, volume 52 of Graduate Texts in Mathematics. Springer, 1977.

[Hwa01] J.-M. Hwang. Geometry of Minimial Rational Curves on Fano Manifolds ICTP Lecture Notes Series Volume VI (ISBN 92-95003-09-8), December 2001. Available on the internet at

         http://www.ictp.trieste.it/∼pub_off/lectures/vol6.html

[HM02] J.-M. Hwang and N. Mok. Automorphism groups of the spaces of minimal rational curves on Fano manifolds of Picard number 1 Preprint, to appear.

[Keb01a] S. Kebekus. Rationale Kurven auf projektiven Mannigfaltigkeiten (German). Habilitationsschrift, February 2001. Available from the author’s home page at

         http://btm8x5.mat.uni-bayreuth.de/∼kebekus

[Keb01b] S. Kebekus. Lines on Contact Manifolds II. LANL-Preprint math.AG/0103208, 2001.

[Keb01c] S. Kebekus. Lines on contact manifolds. J. Reine Angew. Math, 539:167–177, October 2001.

[Keb02a] S. Kebekus. Families of singular rational curves. J. Alg. Geom., 11:245–256, 2002.

[Keb02b] S. Kebekus. Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron. In I. Bauer, F. Catanese, Y. Kawamata, T. Peternell, and Y.-T. Siu, editors, Complex Geometry, Collection of Papers dedicated to Hans Grauert, pages 147–156. Springer, 2002.

[KMM92] J. Kollár, Y. Miyaoka, and S. Mori. Rational Connectedness and Boundedness of Fano Manifolds. J. Diff. Geom., 36:765–769, 1992.

[Kol96] J. Kollár. Rational Curves on Algebraic Varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge. Springer, 2nd edition, 1996.

Stefan Kebekus, Mathematik VIII, Universität Bayreuth, 95440 Bayreuth, Germany

   E-mail address: stefan.kebekus@uni-bayreuth.de
   URL: http://btm8x5.mat.uni-bayreuth.de/∼kebekus

Sándor Kovács, University of Washington, Department of Mathematics, Box 354350, Seattle, WA 98195, U.S.A.

   E-mail address: kovacs@math.washington.edu
   URL: http://www.math.washington.edu/∼kovacs