Families of rationally connected varieties

By Tom Graber, Joe Harris, and Jason Starr


We prove that every one-parameter family of complex rationally connected varieties has a section.

1. Introduction

1.1. Statement of results

We will work throughout over the complex numbers, so that the results here apply over any algebraically closed field of characteristic 0.

Recall that a proper variety is said to be rationally connected if two general points are contained in the image of a map . This is clearly a birationally invariant property. When is smooth, this turns out to be equivalent to the a priori weaker condition that two general points can be joined by a chain of rational curves, and also to the a priori stronger condition that for any finite subset there is a map whose image contains and such that is an ample bundle.

Rationally connected varieties form an important class of varieties. In dimensions 1 and 2 rational connectivity coincides with rationality, but the two notions diverge in higher dimensions and in virtually every respect the class of rationally connected varieties is better behaved. For example, the condition of rational connectivity is both open and closed in smooth proper families; there are geometric criteria for rational connectivity (e.g., any smooth projective variety with negative canonical bundle is rationally connected, so we know in particular that a smooth hypersurface of degree will be rationally connected if and only if ), and there are, at least conjecturally, numerical criteria for rational connectivity (see Conjecture 1.6 below). In this paper we will prove a conjecture of Kollár, Miyaoka and Mori that represents one more basic property of rational connectivity (also one not shared by rationality): if is a morphism with rationally connected image and fibers, then the domain is rationally connected as well. This will be a corollary of our main theorem:

Theorem 1.1.

Let be a proper morphism of complex varieties with a smooth curve. If the general fiber of is rationally connected, then has a section.

Since this is really a statement about the birational equivalence class of the morphism , we can restate it in the equivalent form

Theorem 1.2.

If is the function field of a curve over , then any rationally connected variety defined over has a -rational point.

In this form, the theorem directly generalizes Tsen’s theorem, which is exactly this statement for a smooth hypersurface of degree in projective space (or more generally a smooth complete intersection in projective space with negative canonical bundle; cf. Reference K, Theorem IV.6.5). It would be interesting to know if in fact rationally connected varieties over other fields necessarily have rational points.

As we indicated, one basic corollary of our main theorem is

Corollary 1.3.

Let be any dominant morphism of complex varieties. If and the general fiber of are rationally connected, then is rationally connected.


Since we work over , we can assume that and are smooth projective varieties. Let and be general points of . We can find a map whose image contains and ; let be the pullback of by . By Theorem 1.1, there is a section of over . We can then connect to by a chain of rational curves in in three stages: connect to the point of intersection of with the fiber of through by a rational curve; connect to by ; and connect to by a rational curve in .

There is a further corollary of Theorem 1.1 based on a construction of Campana and Kollár–Miyaoka–Mori: the maximal rationally connected fibration associated to a variety (see Reference Ca, Reference K or Reference KMM). Briefly, the maximal rationally connected fibration associates to a variety a (birational isomorphism class of) variety and a rational map with the properties that

the fibers of are rationally connected; and conversely

almost all the rational curves in lie in fibers of : for a very general point any rational curve in meeting lies in .

The variety and morphism are unique up to birational isomorphism and are called the mrc quotient and mrc fibration of , respectively. They measure the failure of to be rationally connected: if is rationally connected, is a point, while if is not uniruled, we have . As observed in Reference K, IV.5.6.3, we have the following corollary:

Corollary 1.4.

Let be any variety and its maximal rationally connected fibration. Then is not uniruled.


Suppose that were uniruled, so that for a general point we could find a map whose image contains . By Corollary 1.3, the pullback of by will be rationally connected, which means that every point of the fiber will lie on a rational curve not contained in , contradicting the second defining property of mrc fibrations.

There are conjectured numerical criteria for a variety to be either uniruled or rationally connected. They are

Conjecture 1.5.

Let be a smooth projective variety. Then is uniruled if and only if for all .

Conjecture 1.6.

Let be a smooth projective variety. Then is rationally connected if and only if for all .

For each of these conjectures, the “only if ” part is known and straightforward to prove; the “if ” part represents a very difficult open problem (see for example Reference K, IV.1.12 and IV.3.8.1). As another consequence of our main theorem, we have an implication:

Corollary 1.7.

Conjecture 1.5 implies Conjecture 1.6


This is proved in Reference K, IV.5.7, assuming Theorem 1.1; for completeness we remind the reader of the proof. Let be any smooth projective variety that is not rationally connected; assuming the statement of Conjecture 1.5, we want to show that for some . Let be the mrc fibration of . By hypothesis has dimension , and by Corollary 1.4 is not uniruled. If we assume Conjecture 1.5, then we must have a nonzero section for some . But the line bundle is a summand of the tensor power , so we can view as a global section of that sheaf; pulling it back via , we get a nonzero global section of .

2. Preliminary definitions and constructions

We will be dealing with morphisms satisfying a number of hypotheses, which we collect here for future reference. First of all, in proving Theorem 1.1, we are free to assume that is projective and that is smooth and projective by applying resolution of singularities and Chow’s Lemma to . For the bulk of this paper we will deal with the case ; we will show in section 3.2 below both that the statement for implies the full Theorem 1.1 and, as well, how to modify the argument that follows to apply to general .

Hypothesis 2.1.

is a nonconstant morphism of smooth connected projective varieties over , with . For general , the fiber is rationally connected.

It is necessary to work with a class of maps of curves more general than embeddings. We choose to work with Kontsevich’s stable maps.

Definition 2.2.

A pair of a morphism from a connected, complete, at-worst-nodal curve to , and an ordered set of closed points, called marked points, of is a stable map if


the marked points are all distinct,


the marked points are contained in the smooth locus of , and


for every irreducible component of with arithmetic genus 0 on which is constant, there are at least three special points (nodes or marked points), and for every irreducible component of with arithmetic genus 1 on which is constant, there is at least one special point.

Given a numerical equivalence class , the intersection number is defined in the usual way. Thus defines a class in . For a fixed nonnegative integer , a fixed nonnegative integer , and a class , the Kontsevich moduli stack is defined to be the Deligne-Mumford stack which parametrizes flat families of stable maps such that has arithmetic genus and the class equals . We denote by the coarse moduli space of . For definitions and results about Kontsevich stable maps, the reader is referred to Reference FPReference BM.

Now suppose we have a class having intersection number with a fiber of the map . By Reference BM, Theorem 3.6 we have then a natural morphism

defined by composing a map with and collapsing components of as necessary to make the composition stable.

Definition 2.3.

Let be a morphism satisfying Hypothesis 2.1, and let be a stable map from an irreducible nodal curve of genus to with class . We say that is flexible relative to if the map is dominant at the point ; that is, if any neighborhood of in dominates a neighborhood of in .

Now, it is a classical fact that the variety has a unique irreducible component whose general member corresponds to the map with a smooth curve (see for example Reference C and Reference H, and Reference F, Prop 1.5 for a modern treatment). Since the map is proper, it follows that if admits a flexible curve, then will be surjective onto this component. Moreover, this component contains points corresponding to maps with the property that every irreducible component of on which is nonconstant maps isomorphically via to . (For example, we could simply start with disjoint copies of (with mapping each isomorphically to ) and identify pairs of points on the , each pair lying over the same point of . It is easy to check that such a morphism can be smoothed.)

Proposition 2.4.

If is a morphism satisfying Hypothesis 2.1 and , a flexible stable map, then has a section.

Our goal in what follows, accordingly, will be to construct a flexible curve for an arbitrary satisfying Hypothesis 2.1.

2.1. The first construction

To manufacture our flexible curve, we apply two basic constructions, which we describe here. (These constructions, especially the first, are pretty standard; see for example section II.7 of Reference K.) We start with a basic lemma:

Lemma 2.5.

Let be a smooth curve and any vector bundle on ; let be any positive integer. Let be general points and a general one-dimensional subspace of the fiber of at ; let be the sheaf of rational sections of having at most a simple pole at in the direction and regular elsewhere. For sufficiently large we will have

for any points .


To start with, we will prove simply that . Since this is an open condition, it will suffice to exhibit a particular choice of points and subspaces that works. Denoting the rank of by , we take divisible by and choose points . We then specialize to the case

and so on. In this case we have , which we know has vanishing higher cohomology for sufficiently large .

Given this, the statement of the lemma follows: to begin with, choose any points . Applying the argument thus far to the bundle , we find that for sufficiently large we will have

But now for any points we have

for some effective divisor on . It follows then that

It is well known that given an embedded curve which is at-worst-nodal and given an irreducible component which intersects in the nodes , the restriction (as a sheaf) consists of rational sections of with at worst simple poles at each whose normal direction is determined by the other branch of through . We will need a version of this result when the embedded curves are replaced by stable maps to .

For an embedded LCI curve , the space of first-order deformations and the obstruction group are given by and , respectively. Suppose given an unmarked stable map . By Reference BF and Reference B, the space of first-order deformations of the stable map and the obstruction group are given by the hypercohomology groups

where is the complex

We will only need to use the deformation theory of stable maps when the stable map satisfies the additional hypothesis that the unramified locus contains all nodes of (in particular there are no irreducible components on which is constant). In this case is quasi-isomorphic to a complex where is a coherent sheaf, the normal sheaf of , whose restriction to is isomorphic to the dual of the kernel of .

Suppose and are stable maps such that


the unramified locus of (resp. ) contains all nodes of as well as all marked points,


for every , , and


for every , the tangent lines are distinct.

By Reference BM, Theorem 3.6, there is a unique stable map where is the connected sum of and with identified to , and such that , . Condition (3) above ensures that the points are in the unramified locus of .

Lemma 2.6.

Let be a smooth projective variety and let and let be as above. Let be the stable map obtained by gluing each to . The normal sheaf of the map is locally free near each of the nodes of and we have an inclusion of sheaves

identifying with the sheaf of rational sections of having at most a simple pole at each in the normal direction determined by . Moreover, if is a first-order deformation of corresponding to a global section , then smooths the node of at if and only if the restriction of to a neighborhood of in is not in the image of .

As this lemma concerns only the local behavior near the nodes, it follows immediately from the analogous statement for embedded curves.

Hypothesis 2.7.

Suppose is a morphism satisfying our basic Hypothesis 2.1. An unmarked stable map satisfies our second hypotheses if is smooth and is surjective.

Suppose satisfies our second hypotheses and has genus . For a general point , let be the fiber of through . By hypothesis, is a smooth, rationally connected variety, so that we can find a stable map from a smooth rational curve and a point (at which is unramified) such that , such that the image of in is arbitrary, and such that is an ample bundle on .

Choose a large number of general points , and for each let be a stable map as in the last paragraph. Let denote the stable map obtained by gluing each to by identifying and . Combining the preceding two lemmas, we see that for sufficiently large, the normal sheaf will be generated by its global sections; in particular, by Lemma 2.6 there will be a smooth deformation of (i.e., a general stable map which is a member of a -parameter family of stable maps specializing to ). Moreover, for any given we can choose the number large enough to ensure that

for some points ; it follows that for some and hence that

for any points on .

The process of taking a stable map which satisfies our second hypotheses, attaching rational curves in fibers and smoothing as above to get a new stable map with smooth domain, is our first construction. It has the properties that:


also satisfies our second hypotheses,


the genus of the new curve is the same as the genus of the curve we started with,


the degree of is the same as the degree of ,


the branch divisor of is a small deformation of the branch divisor of , and again,


for any points we have .

Here is one application of this construction. Suppose we have a stable map satisfying our second hypotheses such that the projection is simply branched — that is, the branch divisor of consists of distinct points in — and such that each ramification point of maps to a smooth point of the fiber . Applying our first construction with , we arrive at another stable map satisfying our hypotheses which is again simply branched over , with all ramification points mapping to smooth points of fibers of . But now the condition that applied to the ramification points of the map says that if we pick a normal vector at each ramification point of , we can find a global section of the normal sheaf with value at . Moreover, since ramification occurs at smooth points of fibers of , for any tangent vectors to at the image points we can find tangent vectors with . It follows that as we deform the stable map , the branch points of move independently. A general deformation of thus yields a general deformation of — in other words, the stable map is flexible. We thus make the

Definition 2.8.

Let be as in Hypothesis 2.1, and let be a stable map as in Hypothesis 2.7 such that the projection is simply branched. If each ramification point of maps to a smooth point of the fiber containing it, we will say the stable map is pre-flexible.

In these terms, we have established the

Lemma 2.9.

Let be as in Hypothesis 2.1. If admits a pre-flexible stable map, the map has a section.

2.2. The second construction

Our second construction is a very minor modification of the first. Given a family as in Hypothesis 2.1 and a stable map satisfying Hypothesis 2.7, we pick a general fiber