We prove that every one-parameter family of complex rationally connected varieties has a section.
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:
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 ,
If is the function field of a curve over then any rationally connected variety , defined over has a point. -rational
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. ReferenceK, 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
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 ReferenceCa, ReferenceK or ReferenceKMM). 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 .ReferenceK, IV.5.6.3, we have the following corollary:
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
Let be a smooth projective variety. Then is uniruled if and only if for all .
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 ReferenceK, IV.1.12 and IV.3.8.1). As another consequence of our main theorem, we have an implication:
This is proved in ReferenceK, 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 ,.
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 .
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.
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 .ReferenceFP, ReferenceBM.
Now suppose we have a class having intersection number with a fiber of the map By .ReferenceBM, 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.
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 ReferenceC and ReferenceH, and ReferenceF, 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.) .
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.
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 ReferenceK.) We start with a basic lemma:
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 .ReferenceBF and ReferenceB, 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 ReferenceBM, 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 .
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.
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 family of stable maps specializing to -parameter 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
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
Let be as in Hypothesis 2.1. If admits a pre-flexible stable map, the map has a section.
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 of and two points such that We then pick a stable map . with smooth, rational domain and two points such that and such that is an ample bundle.
We also pick a large number of other general points and stable maps in the corresponding fibers and points such that such that , are ample and such that is a general line in (just as in the first construction). Finally, we let be the stable map obtained by gluing and by identifying with , with and for each identifying with As in the first construction, for . large enough, there will be deformations of with smooth domain, and we choose a general smooth deformation of (i.e., a general stable map which is a member of a family of stable maps specializing to -parameter This process, starting with the stable map ). and arriving at the new stable map is our second construction. It has the properties that ,
satisfies Hypothesis 2.7,
the degree of is the same as the degree of ,
the genus of the new curve is one greater than the genus of the curve we started with,
for any points we have and ,
the branch divisor of has two new points: it consists of a small deformation of the branch divisor of together with a pair of simple branch points , near each having as monodromy the transposition exchanging the sheets of , near and .
In effect, we have simply introduced two new simple branch points to the cover with assigned (though necessarily equal) monodromy. Note that we can apply this construction repeatedly, to introduce any number of (pairs of) additional branch points with assigned (simple) monodromy; or we could carry out a more general construction with a number of curves ,.
We are now more than amply equipped to prove the theorem. We start with a morphism as in Hypothesis 2.1. To begin with, by hypothesis is projective; embed in a projective space and take the intersection with general hyperplanes to arrive at a smooth curve The inclusion . satisfies Hypothesis 2.7 and is the stable map we start with.
What do and the associated map look like? To answer this, start with the simplest case: suppose that the fibers of do not have multiple components, or in other words that the singular locus of the map has codimension 2 in In this case we are done: . misses altogether, so that all ramification of occurs at smooth points of fibers; and simple dimension counts show we can choose so that the branching of is simple. In other words, is pre-flexible already.
The problems start if has multiple components of fibers. If is such a component, then each point will be a ramification point of and no deformation of , will move the corresponding branch point The curve . cannot be flexible. And of course it is worse if has a multiple (that is, everywhere-nonreduced) fiber: in that case cannot possibly have a section (a posteriori we will see this cannot occur).
To keep track of such points, let be the locus of points such that the fiber has a multiple component. Outside of the map , is simply branched, and all ramification occurs at smooth points of fibers of .
Now here is what we are going to do. First, pick a base point and draw a cut system: that is, a collection of real arcs joining , to the branch points of disjoint except at , The inverse image in . of the complement of these arcs is simply disjoint copies of call the set of sheets ; (or, if you prefer, label them with the integers 1 through Now, for each point ). denote the monodromy around the point , by and express this permutation of , as a product of transpositions:
so that in other words
is the identity. For future reference, let We will proceed in three stages. .
Stage 1: We use our second construction to produce a new stable map such that is unramified at and such that in a neighborhood of each point we have new pairs of simple branch points of say , with the monodromy around , and equal to Note that . will have genus and that the branch divisor of the projection will be the union of of a small deformation , of and of all the points , and In particular we can find disjoint discs . with , containing the points and so that the monodromy around the boundary , of is trivial.
Now, for any fixed integer this construction can be carried out so that the stable map has the property that for any points Here we want to choose .
so that there are global sections of the normal sheaf with arbitrarily assigned values on the ramification points of over and the points and This means in particular that we can deform the stable map . so as to deform the branch points of outside of independently. What we will do, then, is
Stage 2: We will vary so as to keep all the branch points and all the points fixed; and for each specialize all the branch points to within the disc .
This is illustrated in Figure 1. To say this more precisely, let be the class of the stable map and consider the maps ,
with the second map assigning to a stable map its branch divisor (we only need the branch morphism as a set map, but in fact it is a regular morphism ReferenceFaP). What we are saying is, starting at the branch divisor
of the map draw an analytic arc , in the subvariety
tending to the point
Since the image of the composition
contains we can find an arc , in that maps onto with , the inclusion .
Stage 3: Let be the limit, in of the family of curves constructed in Stage 2; that is, the point of the arc , over Let . be the normalization of any irreducible component of on which the composition is nonconstant (that is, whose image is not contained in a fiber), and let be the stable map where is the restriction of to In particular, . is a stable map satisfying Hypothesis 2.7.
By construction, the composition is unramified over a neighborhood of the monodromy around the boundary : of each disc is trivial, and it can be branched over at most one point inside so it cannot be branched at all over , Indeed, it is (at most) simply branched over each point of . and each point and unramified elsewhere. Moreover, since we can carry out the specialization of , above with the entire fiber of over the points of and the fixed, the ramification of over these points will occur at smooth points of the corresponding fibers of In other words, the map . is pre-flexible, and we are done.
As we indicated at the outset, there are two straightforward ways of extending this result to the case of arbitrary curves .
For one thing, virtually all of the argument we have made goes over without change to the case of base curves of any genus The one exception to this is the statement that the space . of stable maps of degree from curves of genus to has a unique irreducible component whose general member corresponds to a flat map This is false in general—consider for example the case . of unramified covers. It is true, however, if we restrict ourselves to the case (that is, we have a large number of branch points) and look only at covers whose monodromy is the full symmetric group ReferenceGHS, Theorem 1. Given this fact and observing that our second construction allows us to increase the number of branch points of our covers arbitrarily, the theorem can be proved for general just as it is proved above for .
Alternatively, Johan de Jong showed us a simple way to deduce the theorem for general from the case alone. We argue as follows: given a map with rationally connected general fiber, we choose any map expressing as a branched cover of We can then form the “norm” of . this is the (birational isomorphism class of) variety : whose fiber over a general point is the product
Since the product of rationally connected varieties is again rationally connected, it follows from the case of the theorem that has a rational section, and hence so does .
We would like to thank Johan de Jong, János Kollár and Barry Mazur for many conversations, which were of tremendous help to us. We would also like to thank Olivier Debarre, Vyacheslav Shokurov and the referee for useful comments on preliminary versions of this work.