Families of rationally connected varieties
We prove that every one-parameter family of complex rationally connected varieties has a section.
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:
Since this is really a statement about the birational equivalence class of the morphism we can restate it in the equivalent form ,
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
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:
There are conjectured numerical criteria for a variety to be either uniruled or rationally connected. They are
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:
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 .
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.
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 FP, Reference 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.
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.) .
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:
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 .
As this lemma concerns only the local behavior near the nodes, it follows immediately from the analogous statement for embedded curves.
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
In these terms, we have established the