American Mathematical Society

Families of rationally connected varieties

By Tom Graber, Joe Harris, Jason Starr

Abstract

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 upper X is said to be rationally connected if two general points p comma q element-of upper X are contained in the image of a map g colon double-struck upper P Superscript 1 Baseline right-arrow upper X . This is clearly a birationally invariant property. When upper X 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 normal upper Gamma subset-of upper X there is a map g colon double-struck upper P Superscript 1 Baseline right-arrow upper X whose image contains normal upper Gamma and such that g Superscript asterisk Baseline upper T Subscript upper X 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 upper X subset-of double-struck upper P Superscript n of degree d will be rationally connected if and only if d less-than-or-equal-to n ), 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 upper X right-arrow upper Y is a morphism with rationally connected image and fibers, then the domain upper X is rationally connected as well. This will be a corollary of our main theorem:

Theorem 1.1

Let f colon upper X right-arrow upper B be a proper morphism of complex varieties with upper B a smooth curve. If the general fiber of f is rationally connected, then f has a section.

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

Theorem 1.2

If upper K is the function field of a curve over double-struck upper C , then any rationally connected variety upper X defined over upper K has a upper K -rational point.

In this form, the theorem directly generalizes Tsen’s theorem, which is exactly this statement for upper X a smooth hypersurface of degree d less-than-or-equal-to n in projective space double-struck upper P Superscript n (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 upper C 1 fields necessarily have rational points.

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

Corollary 1.3

Let f colon upper X right-arrow upper Y be any dominant morphism of complex varieties. If upper Y and the general fiber of f are rationally connected, then upper X is rationally connected.

Proof.

Since we work over double-struck upper C , we can assume that upper X and upper Y are smooth projective varieties. Let p and q be general points of upper X . We can find a map g colon double-struck upper P Superscript 1 Baseline right-arrow upper Y whose image contains f left-parenthesis p right-parenthesis and f left-parenthesis q right-parenthesis ; let upper X prime equals double-struck upper P Superscript 1 Baseline times Subscript upper Y Baseline upper X be the pullback of f prime by g . By Theorem 1.1, there is a section upper D of upper X prime over double-struck upper P Superscript 1 . We can then connect p to q by a chain of rational curves in upper X prime in three stages: connect p to the point upper D intersection upper X Subscript p of intersection of upper D with the fiber upper X Subscript p of f through p by a rational curve; connect upper D intersection upper X Subscript p to upper D intersection upper X Subscript q by upper D ; and connect upper D intersection upper X Subscript q to q by a rational curve in upper X Subscript q .

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 upper X (see ReferenceCa, ReferenceK or ReferenceKMM). Briefly, the maximal rationally connected fibration associates to a variety upper X a (birational isomorphism class of) variety upper Z and a rational map phi colon upper X right-arrow upper Z with the properties that

the fibers upper X Subscript z of phi are rationally connected; and conversely

almost all the rational curves in upper X lie in fibers of phi : for a very general point z element-of upper Z any rational curve in upper X meeting upper X Subscript z lies in upper X Subscript z .

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

Corollary 1.4

Let upper X be any variety and phi colon upper X right-arrow upper Z its maximal rationally connected fibration. Then upper Z is not uniruled.

Proof.

Suppose that upper Z were uniruled, so that for a general point z element-of upper Z we could find a map g colon double-struck upper P Superscript 1 Baseline right-arrow upper Z whose image contains z . By Corollary 1.3, the pullback upper X prime equals double-struck upper P Superscript 1 Baseline times Subscript upper Z Baseline upper X of phi by g will be rationally connected, which means that every point of the fiber upper X Subscript z will lie on a rational curve not contained in upper X Subscript z , contradicting the second defining property of mrc fibrations.

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

Conjecture 1.5

Let upper X be a smooth projective variety. Then upper X is uniruled if and only if upper H Superscript 0 Baseline left-parenthesis upper X comma upper K Subscript upper X Superscript m Baseline right-parenthesis equals 0 for all m greater-than 0 .

Conjecture 1.6

Let upper X be a smooth projective variety. Then upper X is rationally connected if and only if upper H Superscript 0 Baseline left-parenthesis upper X comma left-parenthesis normal upper Omega Subscript upper X Superscript 1 Baseline right-parenthesis Superscript circled-times m Baseline right-parenthesis equals 0 for all m greater-than 0 .

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:

Corollary 1.7

Conjecture 1.5 implies Conjecture 1.6

Proof.

This is proved in ReferenceK, IV.5.7, assuming Theorem 1.1; for completeness we remind the reader of the proof. Let upper X be any smooth projective variety that is not rationally connected; assuming the statement of Conjecture 1.5, we want to show that upper H Superscript 0 Baseline left-parenthesis upper X comma left-parenthesis normal upper Omega Subscript upper X Superscript 1 Baseline right-parenthesis Superscript circled-times m Baseline right-parenthesis not-equals 0 for some m greater-than 0 . Let phi colon upper X right-arrow upper Z be the mrc fibration of upper X . By hypothesis upper Z has dimension n greater-than 0 , and by Corollary 1.4 upper Z is not uniruled. If we assume Conjecture 1.5, then we must have a nonzero section sigma element-of upper H Superscript 0 Baseline left-parenthesis upper Z comma upper K Subscript upper Z Superscript m Baseline right-parenthesis for some m greater-than 0 . But the line bundle upper K Subscript upper Z Superscript m is a summand of the tensor power left-parenthesis normal upper Omega Subscript upper Z Superscript 1 Baseline right-parenthesis Superscript circled-times n m , so we can view sigma as a global section of that sheaf; pulling it back via phi , we get a nonzero global section of left-parenthesis normal upper Omega Subscript upper X Superscript 1 Baseline right-parenthesis Superscript circled-times n m .

2. Preliminary definitions and constructions

We will be dealing with morphisms pi colon upper X right-arrow upper B 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 upper B is projective and that upper X is smooth and projective by applying resolution of singularities and Chow’s Lemma to upper X . For the bulk of this paper we will deal with the case upper B approximately-equals double-struck upper P Superscript 1 ; we will show in section 3.2 below both that the statement for upper B approximately-equals double-struck upper P Superscript 1 implies the full Theorem 1.1 and, as well, how to modify the argument that follows to apply to general upper B .

Hypothesis 2.1

pi colon upper X right-arrow upper B is a nonconstant morphism of smooth connected projective varieties over double-struck upper C , with upper B approximately-equals double-struck upper P Superscript 1 . For general b element-of upper B , the fiber upper X Subscript b Baseline equals pi Superscript negative 1 Baseline left-parenthesis b right-parenthesis is rationally connected.

It is necessary to work with a class of maps of curves g colon upper C right-arrow upper X more general than embeddings. We choose to work with Kontsevich’s stable maps.

Definition 2.2

A pair left-parenthesis f colon upper C right-arrow upper X comma left-parenthesis p 1 comma ellipsis comma p Subscript n Baseline right-parenthesis right-parenthesis of a morphism from a connected, complete, at-worst-nodal curve upper C to upper X , and an ordered set of closed points, called marked points, of upper C is a stable map if

(1)

the marked points p Subscript i are all distinct,

(2)

the marked points p Subscript i are contained in the smooth locus of upper C , and

(3)

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

Given a numerical equivalence class left-bracket upper D right-bracket element-of upper N Superscript 1 Baseline left-parenthesis upper X right-parenthesis , the intersection number f Subscript asterisk Baseline left-bracket upper C right-bracket period left-bracket upper D right-bracket is defined in the usual way. Thus f Subscript asterisk Baseline left-bracket upper C right-bracket defines a class in upper N 1 left-parenthesis upper X right-parenthesis . For a fixed nonnegative integer g , a fixed nonnegative integer n , and a class beta element-of upper N 1 left-parenthesis upper X right-parenthesis , the Kontsevich moduli stack ModifyingAbove script upper M With bar Subscript g comma n Baseline left-parenthesis upper X comma beta right-parenthesis is defined to be the Deligne-Mumford stack which parametrizes flat families of stable maps left-parenthesis f colon upper C right-arrow upper X comma left-parenthesis p 1 comma ellipsis comma p Subscript n Baseline right-parenthesis right-parenthesis such that upper C has arithmetic genus g and the class f Subscript asterisk Baseline left-bracket upper C right-bracket equals beta . We denote by ModifyingAbove upper M With bar Subscript g comma n Baseline left-parenthesis upper X comma beta right-parenthesis the coarse moduli space of ModifyingAbove script upper M With bar Subscript g comma n Baseline left-parenthesis upper X comma beta right-parenthesis . For definitions and results about Kontsevich stable maps, the reader is referred to ReferenceFP, ReferenceBM.

Now suppose we have a class beta element-of upper N 1 left-parenthesis upper X right-parenthesis having intersection number d with a fiber of the map pi . By ReferenceBM, Theorem 3.6 we have then a natural morphism

phi colon ModifyingAbove upper M With bar Subscript g comma 0 Baseline left-parenthesis upper X comma beta right-parenthesis right-arrow ModifyingAbove upper M With bar Subscript g comma 0 Baseline left-parenthesis upper B comma d right-parenthesis

defined by composing a map f colon upper C right-arrow upper X with pi and collapsing components of upper C as necessary to make the composition pi f stable.

Definition 2.3

Let pi colon upper X right-arrow upper B be a morphism satisfying Hypothesis 2.1, and let f colon upper C right-arrow upper X be a stable map from an irreducible nodal curve upper C of genus g to upper X with class f Subscript asterisk Baseline left-bracket upper C right-bracket equals beta . We say that f is flexible relative to pi if the map phi colon ModifyingAbove upper M With bar Subscript g comma 0 Baseline left-parenthesis upper X comma beta right-parenthesis right-arrow ModifyingAbove upper M With bar Subscript g comma 0 Baseline left-parenthesis upper B comma d right-parenthesis is dominant at the point left-bracket f right-bracket element-of ModifyingAbove upper M With bar Subscript g comma 0 Baseline left-parenthesis upper X comma beta right-parenthesis ; that is, if any neighborhood of left-bracket f right-bracket in ModifyingAbove upper M With bar Subscript g comma 0 Baseline left-parenthesis upper X comma beta right-parenthesis dominates a neighborhood of left-bracket pi f right-bracket in ModifyingAbove upper M With bar Subscript g comma 0 Baseline left-parenthesis upper B comma d right-parenthesis .

Now, it is a classical fact that the variety ModifyingAbove upper M With bar Subscript g comma 0 Baseline left-parenthesis upper B comma d right-parenthesis has a unique irreducible component whose general member corresponds to the map f colon upper C right-arrow upper B with upper C a smooth curve (see for example ReferenceC and ReferenceH, and ReferenceF, Prop 1.5 for a modern treatment). Since the map phi colon ModifyingAbove upper M With bar Subscript g comma 0 Baseline left-parenthesis upper X comma beta right-parenthesis right-arrow ModifyingAbove upper M With bar Subscript g comma 0 Baseline left-parenthesis upper B comma d right-parenthesis is proper, it follows that if pi colon upper X right-arrow upper B admits a flexible curve, then phi will be surjective onto this component. Moreover, this component contains points left-bracket f right-bracket corresponding to maps f colon upper C right-arrow upper B with the property that every irreducible component of upper C on which f is nonconstant maps isomorphically via f to upper B . (For example, we could simply start with d disjoint copies upper C 1 comma ellipsis comma upper C Subscript d Baseline of upper B (with f mapping each isomorphically to upper B ) and identify d plus g minus 1 pairs of points on the upper C Subscript i , each pair lying over the same point of upper B . It is easy to check that such a morphism can be smoothed.)

Proposition 2.4

If pi colon upper X right-arrow upper B is a morphism satisfying Hypothesis 2.1 and f colon upper C right-arrow upper X , a flexible stable map, then pi has a section.

Our goal in what follows, accordingly, will be to construct a flexible curve f colon upper C right-arrow upper X for an arbitrary pi colon upper X right-arrow upper B 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 ReferenceK.) We start with a basic lemma:

Lemma 2.5

Let upper C be a smooth curve and upper E any vector bundle on upper C ; let n be any positive integer. Let p 1 comma ellipsis comma p Subscript upper N Baseline element-of upper C be general points and xi Subscript i Baseline subset-of upper E Subscript p Sub Subscript i a general one-dimensional subspace of the fiber of upper E at p Subscript i ; let upper E prime be the sheaf of rational sections of upper E having at most a simple pole at p Subscript i in the direction xi Subscript i and regular elsewhere. For upper N sufficiently large we will have

upper H Superscript 1 Baseline left-parenthesis upper C comma upper E prime left-parenthesis minus q 1 minus dot dot dot minus q Subscript n Baseline right-parenthesis right-parenthesis equals 0

for any n points q 1 comma ellipsis comma q Subscript n Baseline element-of upper C .

Proof.

To start with, we will prove simply that upper H Superscript 1 Baseline left-parenthesis upper C comma upper E Superscript prime Baseline right-parenthesis equals 0 . Since this is an open condition, it will suffice to exhibit a particular choice of points p Subscript i and subspaces xi Subscript i that works. Denoting the rank of upper E by r , we take upper N equals m r divisible by r and choose m points t 1 comma ellipsis comma t Subscript m Baseline element-of upper C . We then specialize to the case

StartLayout 1st Row 1st Column p 1 equals dot dot dot equals p Subscript r Baseline 2nd Column equals t 1 semicolon 3rd Column xi 1 comma ellipsis comma xi Subscript r Baseline 4th Column spanning upper E Subscript t 1 Baseline comma 2nd Row 1st Column p Subscript r plus 1 Baseline equals dot dot dot equals p Subscript 2 r Baseline 2nd Column equals t 2 semicolon 3rd Column xi Subscript r plus 1 Baseline comma ellipsis comma xi Subscript 2 r Baseline 4th Column spanning upper E Subscript t 2 Baseline comma EndLayout

and so on. In this case we have upper E prime equals upper E left-parenthesis t 1 plus dot dot dot plus t Subscript m Baseline right-parenthesis , which we know has vanishing higher cohomology for sufficiently large m .

Given this, the statement of the lemma follows: to begin with, choose any g plus n points r 1 comma ellipsis comma r Subscript g plus n Baseline element-of upper C . Applying the argument thus far to the bundle upper E left-parenthesis minus r 1 minus dot dot dot minus r Subscript g plus n Baseline right-parenthesis , we find that for upper N sufficiently large we will have

upper H Superscript 1 Baseline left-parenthesis upper C comma upper E prime left-parenthesis minus r 1 minus dot dot dot minus r Subscript g plus n Baseline right-parenthesis right-parenthesis equals 0 period

But now for any points q 1 comma ellipsis comma q Subscript n Baseline element-of upper C we have

q 1 plus dot dot dot plus q Subscript n Baseline equals r 1 plus dot dot dot plus r Subscript g plus n Baseline minus upper D

for some effective divisor upper D on upper C . It follows then that

StartLayout 1st Row 1st Column h Superscript 1 Baseline left-parenthesis upper C comma upper E prime left-parenthesis minus q 1 minus dot dot dot minus q Subscript n Baseline right-parenthesis right-parenthesis 2nd Column equals h Superscript 1 Baseline left-parenthesis upper C comma upper E prime left-parenthesis minus r 1 minus dot dot dot minus r Subscript g plus n Baseline right-parenthesis left-parenthesis upper D right-parenthesis right-parenthesis 2nd Row 1st Column Blank 2nd Column h Superscript 1 Baseline left-parenthesis upper C comma upper E prime left-parenthesis minus r 1 minus dot dot dot minus r Subscript g plus n Baseline right-parenthesis right-parenthesis 3rd Row 1st Column Blank 2nd Column equals 0 period EndLayout

It is well known that given an embedded curve upper D subset-of upper X which is at-worst-nodal and given an irreducible component upper C subset-of upper D which intersects ModifyingAbove upper D minus upper C With bar in the nodes p 1 comma ellipsis comma p Subscript delta Baseline , the restriction upper N Subscript upper D slash upper X Baseline vertical-bar Subscript upper C Baseline (as a sheaf) consists of rational sections of upper N Subscript upper C slash upper X with at worst simple poles at each p Subscript i whose normal direction is determined by the other branch of upper D through p Subscript i . We will need a version of this result when the embedded curves are replaced by stable maps to upper X .

For an embedded LCI curve upper C subset-of upper X , the space of first-order deformations and the obstruction group are given by upper H Superscript 0 Baseline left-parenthesis upper C comma upper N Subscript upper C slash upper X Baseline right-parenthesis and upper H Superscript 1 Baseline left-parenthesis upper C comma upper N Subscript upper C slash upper X Baseline right-parenthesis , respectively. Suppose given an unmarked stable map f colon upper C right-arrow upper X . By ReferenceBF and ReferenceB, the space of first-order deformations of the stable map and the obstruction group are given by the hypercohomology groups

Def left-parenthesis f right-parenthesis equals double-struck upper H Superscript 1 Baseline left-parenthesis upper C comma double-struck upper R italic hom Subscript script upper O Sub Subscript upper C Subscript Baseline left-parenthesis normal upper Omega Subscript f Superscript dot Baseline comma script upper O Subscript upper C Baseline right-parenthesis right-parenthesis comma

Obs left-parenthesis f right-parenthesis equals double-struck upper H squared left-parenthesis upper C comma double-struck upper R italic hom Subscript script upper O Sub Subscript upper C Subscript Baseline left-parenthesis normal upper Omega Subscript f Superscript dot Baseline comma script upper O Subscript upper C Baseline right-parenthesis right-parenthesis comma where normal upper Omega Subscript f Superscript dot is the complex

StartLayout 1st Row with Label left-parenthesis 1 right-parenthesis EndLabel StartLayout 1st Row 1st Column negative 1 2nd Column Blank 3rd Column 0 2nd Row 1st Column f Superscript asterisk Baseline normal upper Omega Subscript upper X 2nd Column Blank 3rd Column right-arrow Overscript d f Superscript dagger Endscripts 4th Column normal upper Omega Subscript upper C Baseline period EndLayout EndLayout

We will only need to use the deformation theory of stable maps when the stable map satisfies the additional hypothesis that the unramified locus upper U subset-of upper C contains all nodes of upper C (in particular there are no irreducible components on which f is constant). In this case double-struck upper R italic hom Subscript script upper O Sub Subscript upper C Baseline left-parenthesis normal upper Omega Subscript f Superscript dot Baseline comma script upper O Subscript upper C Baseline right-parenthesis is quasi-isomorphic to a complex script upper N Subscript f Baseline left-bracket negative 1 right-bracket where script upper N Subscript f is a coherent sheaf, the normal sheaf of f , whose restriction to upper U is isomorphic to the dual of the kernel of f Superscript asterisk Baseline normal upper Omega Subscript upper X Baseline right-arrow normal upper Omega Subscript upper C .

Suppose left-parenthesis f colon upper C right-arrow upper X comma left-parenthesis p 1 comma ellipsis comma p Subscript delta Baseline right-parenthesis right-parenthesis and left-parenthesis f prime colon upper C prime right-arrow upper X comma left-parenthesis p prime 1 comma ellipsis comma p prime Subscript delta right-parenthesis right-parenthesis are stable maps such that

(1)

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

(2)

for every i equals 1 comma ellipsis comma delta , f left-parenthesis p Subscript i Baseline right-parenthesis equals f prime left-parenthesis p prime Subscript i right-parenthesis , and

(3)

for every i equals 1 comma ellipsis comma delta , the tangent lines d f left-parenthesis upper T Subscript upper C comma p Sub Subscript i Subscript Baseline right-parenthesis comma d f prime left-parenthesis upper T Subscript upper C prime comma p prime Sub Subscript i Baseline right-parenthesis subset-of upper T Subscript upper X comma f left-parenthesis p Sub Subscript i Subscript right-parenthesis Baseline are distinct.

By ReferenceBM, Theorem 3.6, there is a unique stable map upper F colon upper D right-arrow upper X where upper D is the connected sum of upper C and upper C prime with p Subscript i identified to p prime Subscript i , and such that upper F vertical-bar Subscript upper C Baseline equals f , upper F vertical-bar Subscript upper C Sub Superscript prime Subscript Baseline equals f prime . Condition (3) above ensures that the points p Subscript i are in the unramified locus of upper F .

Lemma 2.6

Let upper X be a smooth projective variety and let left-parenthesis f colon upper C right-arrow upper X comma left-parenthesis p 1 comma ellipsis comma p Subscript delta Baseline right-parenthesis right-parenthesis and let left-parenthesis f prime colon upper C right-arrow upper X comma left-parenthesis p prime 1 comma ellipsis comma p prime Subscript delta right-parenthesis right-parenthesis be as above. Let left-parenthesis upper F colon upper D right-arrow upper X right-parenthesis be the stable map obtained by gluing each p Subscript i to p prime Subscript i . The normal sheaf of the map script upper N Subscript upper F is locally free near each of the nodes p Subscript i of upper D and we have an inclusion of sheaves

0 right-arrow script upper N Subscript f Baseline right-arrow script upper N Subscript upper F Baseline vertical-bar Subscript upper C Baseline

identifying script upper N Subscript upper F Baseline vertical-bar Subscript upper C Baseline with the sheaf of rational sections of script upper N Subscript f having at most a simple pole at each p Subscript i in the normal direction determined by d f prime left-parenthesis upper T Subscript p prime Sub Subscript i Baseline upper C prime right-parenthesis . Moreover, if left-parenthesis upper F overTilde colon upper D overTilde right-arrow upper X right-parenthesis is a first-order deformation of upper F corresponding to a global section sigma element-of upper H Superscript 0 Baseline left-parenthesis upper D comma script upper N Subscript upper F Baseline right-parenthesis , then upper D overTilde smooths the node of upper D at p Subscript i if and only if the restriction sigma vertical-bar Subscript upper U Baseline of sigma to a neighborhood upper U of p Subscript i in upper C is not in the image of script upper N Subscript f .

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 pi colon upper X right-arrow upper B is a morphism satisfying our basic Hypothesis 2.1. An unmarked stable map f colon upper C right-arrow upper X satisfies our second hypotheses if upper C is smooth and mu colon equals pi ring f is surjective.

Suppose f colon upper C right-arrow upper X satisfies our second hypotheses and upper C has genus g . For a general point p element-of upper C , let upper X Subscript p Baseline equals pi Superscript negative 1 Baseline left-parenthesis pi left-parenthesis f left-parenthesis p right-parenthesis right-parenthesis right-parenthesis be the fiber of pi through f left-parenthesis p right-parenthesis . By hypothesis, upper X Subscript p is a smooth, rationally connected variety, so that we can find a stable map left-parenthesis f prime colon upper C prime subset-of upper X Subscript p Baseline right-parenthesis from a smooth rational curve and a point p prime right-arrow upper C prime (at which f prime is unramified) such that f left-parenthesis p right-parenthesis equals f prime left-parenthesis p prime right-parenthesis , such that the image of d f prime left-parenthesis upper T Subscript p prime Baseline upper C prime right-parenthesis in upper T Subscript f left-parenthesis p right-parenthesis Baseline upper X slash d f left-parenthesis upper T Subscript p Baseline upper C right-parenthesis equals left-parenthesis script upper N Subscript upper C prime slash upper X Baseline right-parenthesis Subscript p is arbitrary, and such that left-parenthesis f prime right-parenthesis Superscript asterisk Baseline upper T upper X Subscript p is an ample bundle on upper C prime .

Choose a large number of general points p 1 comma ellipsis comma p Subscript delta Baseline element-of upper C , and for each i let f prime Subscript i Baseline colon upper C prime Subscript i Baseline subset-of upper X Subscript p Sub Subscript i Subscript Baseline be a stable map as in the last paragraph. Let upper F colon upper D right-arrow upper X denote the stable map obtained by gluing each upper C prime Subscript i to upper C by identifying p Subscript i and p prime Subscript i . Combining the preceding two lemmas, we see that for delta sufficiently large, the normal sheaf script upper N Subscript upper F will be generated by its global sections; in particular, by Lemma 2.6 there will be a smooth deformation f overTilde colon upper C overTilde right-arrow upper X of upper F (i.e., a general stable map which is a member of a 1 -parameter family of stable maps specializing to upper F ). Moreover, for any given n we can choose the number delta large enough to ensure that

upper H Superscript 1 Baseline left-parenthesis upper C comma script upper N Subscript upper F Baseline vertical-bar Subscript upper C Baseline left-parenthesis minus r 1 minus dot dot dot minus r Subscript g plus n Baseline right-parenthesis right-parenthesis equals 0

for some g plus n points r 1 comma ellipsis comma r Subscript g plus n Baseline element-of upper C ; it follows that upper H Superscript 1 Baseline left-parenthesis upper C overTilde comma script upper N Subscript f overTilde Baseline left-parenthesis minus r 1 minus dot dot dot minus r Subscript g plus n Baseline right-parenthesis right-parenthesis equals 0 for some r 1 comma ellipsis comma r Subscript g plus n Baseline element-of upper C overTilde and hence that

upper H Superscript 1 Baseline left-parenthesis upper C overTilde comma script upper N Subscript f overTilde Baseline left-parenthesis minus q 1 minus dot dot dot minus q Subscript n Baseline right-parenthesis right-parenthesis equals 0

for any n points on upper C overTilde .

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

(1)

f overTilde colon upper C overTilde right-arrow upper X also satisfies our second hypotheses,

(2)

the genus g of the new curve upper C overTilde is the same as the genus of the curve upper C we started with,

(3)

the degree d of mu overTilde equals pi ring f overTilde colon upper C overTilde right-arrow upper B is the same as the degree of mu equals pi ring f colon upper C right-arrow upper B ,

(4)

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

(5)

for any n points q 1 comma ellipsis comma q Subscript n Baseline element-of upper C overTilde we have upper H Superscript 1 Baseline left-parenthesis upper C overTilde comma script upper N Subscript f overTilde Baseline left-parenthesis minus q 1 minus dot dot dot minus q Subscript n Baseline right-parenthesis right-parenthesis equals 0 .

Here is one application of this construction. Suppose we have a stable map f colon upper C subset-of upper X satisfying our second hypotheses such that the projection mu is simply branched — that is, the branch divisor of mu consists of 2 d plus 2 g minus 2 distinct points in upper B — and such that each ramification point p element-of upper C of mu maps to a smooth point of the fiber upper X Subscript p . Applying our first construction with n equals 2 d plus 2 g minus 2 , we arrive at another stable map f overTilde colon upper C overTilde right-arrow upper X satisfying our hypotheses which is again simply branched over upper B , with all ramification points mapping to smooth points of fibers of pi . But now the condition that upper H Superscript 1 Baseline left-parenthesis upper C overTilde comma script upper N Subscript f overTilde Baseline left-parenthesis minus q 1 minus dot dot dot minus q Subscript n Baseline right-parenthesis right-parenthesis equals 0 applied to the n equals 2 d plus 2 g minus 2 ramification points of the map mu overTilde equals pi ring f overTilde says that if we pick a normal vector v Subscript i Baseline element-of left-parenthesis script upper N Subscript f overTilde Baseline right-parenthesis Subscript p Sub Subscript i at each ramification point p Subscript i of mu overTilde , we can find a global section of the normal sheaf script upper N Subscript f overTilde with value v Subscript i at p Subscript i . Moreover, since ramification occurs at smooth points of fibers of pi , for any tangent vectors w Subscript i to upper B at the image points ModifyingAbove mu With tilde left-parenthesis p Subscript i Baseline right-parenthesis we can find tangent vectors v Subscript i Baseline element-of left-parenthesis script upper N Subscript f overTilde Baseline right-parenthesis Subscript p Sub Subscript i with d pi left-parenthesis v Subscript i Baseline right-parenthesis equals w Subscript i . It follows that as we deform the stable map f overTilde colon upper C overTilde right-arrow upper X , the branch points of mu overTilde move independently. A general deformation of f overTilde thus yields a general deformation of mu overTilde — in other words, the stable map f overTilde is flexible. We thus make the

Definition 2.8

Let pi colon upper X right-arrow upper B be as in Hypothesis 2.1, and let f colon upper C subset-of upper X be a stable map as in Hypothesis 2.7 such that the projection mu equals pi ring f is simply branched. If each ramification point p element-of upper C of mu maps to a smooth point of the fiber upper X Subscript p containing it, we will say the stable map f colon upper C right-arrow upper X is pre-flexible.

In these terms, we have established the

Lemma 2.9

Let pi colon upper X right-arrow upper B be as in Hypothesis 2.1. If upper X admits a pre-flexible stable map, the map pi has a section.

2.2. The second construction

Our second construction is a very minor modification of the first. Given a family pi colon upper X right-arrow upper B as in Hypothesis 2.1 and a stable map f colon upper C right-arrow upper X satisfying Hypothesis 2.7, we pick a general fiber upper X Subscript b of pi and two points p comma q element-of upper C such that f left-parenthesis p right-parenthesis comma f left-parenthesis q right-parenthesis element-of upper X Subscript b Baseline . We then pick a stable map f prime 0 colon upper C prime 0 right-arrow upper X Subscript b Baseline with smooth, rational domain and two points p prime comma q prime element-of upper C prime 0 such that f prime 0 left-parenthesis p prime right-parenthesis equals f left-parenthesis p right-parenthesis and f prime 0 left-parenthesis q prime right-parenthesis equals f left-parenthesis q right-parenthesis such that left-parenthesis f prime 0 right-parenthesis Superscript asterisk Baseline upper T upper X Subscript b is an ample bundle.

We also pick a large number upper N of other general points p Subscript i Baseline element-of upper C and stable maps f prime Subscript i Baseline colon upper C prime Subscript i Baseline right-arrow upper X Subscript p Sub Subscript i Subscript Baseline in the corresponding fibers and points p prime Subscript i Baseline element-of upper C prime Subscript i such that f prime Subscript i Baseline left-parenthesis p prime Subscript i right-parenthesis equals f left-parenthesis p Subscript i Baseline right-parenthesis , such that left-parenthesis f prime Subscript i right-parenthesis Superscript asterisk Baseline upper T upper X Subscript p Sub Subscript i are ample and such that d f prime Subscript i Baseline left-parenthesis upper T Subscript p prime Sub Subscript i Baseline upper C prime Subscript i right-parenthesis is a general line in upper T Subscript f left-parenthesis p Sub Subscript i Subscript right-parenthesis Baseline upper X Subscript p Sub Subscript i (just as in the first construction). Finally, we let upper F colon upper D right-arrow upper X be the stable map obtained by gluing upper C and upper C prime 0 comma upper C prime 1 comma ellipsis comma upper C prime Subscript upper N by identifying p with p prime , q with q prime and for each i identifying p Subscript i with p prime Subscript i . As in the first construction, for upper N large enough, there will be deformations of upper F colon upper D right-arrow upper X with smooth domain, and we choose f overTilde colon upper C overTilde right-arrow upper X a general smooth deformation of upper D (i.e., a general stable map which is a member of a 1 -parameter family of stable maps specializing to upper F ). This process, starting with the stable map f colon upper C right-arrow upper X and arriving at the new stable map f overTilde colon upper C overTilde right-arrow upper X , is our second construction. It has the properties that

(1)

f overTilde colon upper C overTilde right-arrow upper X satisfies Hypothesis 2.7,

(2)

the degree d of mu overTilde colon equals pi ring f overTilde is the same as the degree of mu colon equals pi ring f ,

(3)

the genus of the new curve upper C overTilde is one greater than the genus of the curve upper C we started with,

(4)

for any n points q 1 comma ellipsis comma q Subscript n Baseline element-of upper C overTilde we have upper H Superscript 1 Baseline left-parenthesis upper C overTilde comma script upper N Subscript f overTilde Baseline left-parenthesis minus q 1 minus dot dot dot minus q Subscript n Baseline right-parenthesis right-parenthesis equals 0 , and

(5)

the branch divisor of mu has two new points: it consists of a small deformation of the branch divisor of mu , together with a pair of simple branch points b prime comma b double-prime element-of upper B near b , each having as monodromy the transposition exchanging the sheets of upper C overTilde near p and q .

In effect, we have simply introduced two new simple branch points to the cover upper C right-arrow upper B , 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 upper C 0 .

3. Proof of the main theorem

3.1. The proof in case upper B equals double-struck upper P Superscript 1

We are now more than amply equipped to prove the theorem. We start with a morphism pi colon upper X right-arrow upper B as in Hypothesis 2.1. To begin with, by hypothesis upper X is projective; embed in a projective space and take the intersection with dimension left-parenthesis upper X right-parenthesis minus 1 general hyperplanes to arrive at a smooth curve upper C subset-of upper X . The inclusion f colon upper C right-arrow with hook upper X satisfies Hypothesis 2.7 and is the stable map we start with.

What do upper C and the associated map mu colon upper C right-arrow with hook upper X right-arrow upper B look like? To answer this, start with the simplest case: suppose that the fibers upper X Subscript b of pi do not have multiple components, or in other words that the singular locus pi Subscript normal s normal i normal n normal g of the map pi has codimension 2 in upper X . In this case we are done: upper C misses pi Subscript normal s normal i normal n normal g altogether, so that all ramification of mu colon upper C right-arrow upper B occurs at smooth points of fibers; and simple dimension counts show we can choose upper C so that the branching of mu is simple. In other words, f colon upper C right-arrow with hook upper X is pre-flexible already.

The problems start if pi has multiple components of fibers. If upper Z subset-of upper X Subscript b is such a component, then each point p element-of upper C intersection upper Z will be a ramification point of mu , and no deformation of upper C will move the corresponding branch point pi left-parenthesis p right-parenthesis element-of upper B . The curve upper C cannot be flexible. And of course it is worse if pi has a multiple (that is, everywhere-nonreduced) fiber: in that case pi cannot possibly have a section (a posteriori we will see this cannot occur).

To keep track of such points, let upper M subset-of upper B be the locus of points such that the fiber upper X Subscript b has a multiple component. Outside of upper M , the map mu colon upper C right-arrow upper B is simply branched, and all ramification occurs at smooth points of fibers of pi .

Now here is what we are going to do. First, pick a base point p 0 element-of upper B , and draw a cut system: that is, a collection of real arcs joining p 0 to the branch points upper M union upper N of mu , disjoint except at p 0 . The inverse image in upper C of the complement upper U of these arcs is simply d disjoint copies of upper U ; call the set of sheets normal upper Gamma (or, if you prefer, label them with the integers 1 through d ). Now, for each point b element-of upper M , denote the monodromy around the point b by sigma Subscript b , and express this permutation of normal upper Gamma as a product of transpositions:

sigma Subscript b Baseline equals tau Subscript b comma 1 Baseline tau Subscript b comma 2 Baseline ellipsis tau Subscript b comma k Sub Subscript b Subscript

so that in other words

tau Subscript b comma k Sub Subscript b Subscript Baseline ellipsis tau Subscript b comma 2 Baseline tau Subscript b comma 1 Baseline sigma Subscript b Baseline equals upper I

is the identity. For future reference, let k equals sigma-summation k Subscript b . We will proceed in three stages.

Stage 1: We use our second construction to produce a new stable map f overTilde colon upper C overTilde right-arrow upper X such that f overTilde is unramified at ModifyingAbove f With tilde Superscript negative 1 Baseline left-parenthesis upper M right-parenthesis and such that in a neighborhood of each point b element-of upper M we have k Subscript b new pairs of simple branch points of mu overTilde equals pi ring f overTilde , say s Subscript b comma i Baseline comma t Subscript b comma i Baseline element-of upper B , with the monodromy around s Subscript b comma i and t Subscript b comma i equal to tau Subscript b comma i . Note that upper C overTilde will have genus g left-parenthesis upper C right-parenthesis plus k and that the branch divisor of the projection mu overTilde will be the union of upper M , of a small deformation upper N overTilde of upper N , and of all the points s Subscript b comma i and t Subscript b comma i . In particular we can find disjoint discs normal upper Delta Subscript b Baseline subset-of upper B , with normal upper Delta Subscript b containing the points b and t Subscript b comma 1 Baseline comma t Subscript b comma 2 Baseline comma ellipsis comma t Subscript b comma k Sub Subscript b Subscript Baseline , so that the monodromy around the boundary partial-differential normal upper Delta Subscript b of normal upper Delta Subscript b is trivial.

Now, for any fixed integer n this construction can be carried out so that the stable map f overTilde has the property that upper H Superscript 1 Baseline left-parenthesis upper C overTilde comma script upper N Subscript f overTilde Baseline left-parenthesis minus q 1 minus dot dot dot minus q Subscript n Baseline right-parenthesis right-parenthesis equals 0 for any n points q Subscript i Baseline element-of upper C overTilde . Here we want to choose

n equals number-sign upper N plus 2 k

so that there are global sections of the normal sheaf script upper N Subscript f overTilde with arbitrarily assigned values on the ramification points of upper C overTilde over upper N and the points s Subscript b comma i and t Subscript b comma i . This means in particular that we can deform the stable map f overTilde colon upper C overTilde right-arrow upper X so as to deform the branch points of mu overTilde outside of upper M independently. What we will do, then, is

Stage 2: We will vary f overTilde colon upper C overTilde right-arrow upper X so as to keep all the branch points b element-of upper N and all the points s Subscript b comma i fixed; and for each b element-of upper M specialize all the branch points t Subscript b comma i to b within the disc normal upper Delta Subscript b .

This is illustrated in Figure 1. To say this more precisely, let beta element-of upper N 1 left-parenthesis upper X right-parenthesis be the class of the stable map f overTilde colon upper C overTilde right-arrow upper X , and consider the maps

ModifyingAbove upper M With bar Subscript g prime comma 0 Baseline left-parenthesis upper X comma beta right-parenthesis long right-arrow ModifyingAbove upper M With bar Subscript g prime comma 0 Baseline left-parenthesis upper B comma d right-parenthesis long right-arrow upper B Subscript 2 d plus 2 g prime minus 2

with the second map assigning to a stable map upper C right-arrow upper B 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

upper D 1 equals upper N overTilde plus sigma-summation s Subscript b comma i Baseline plus sigma-summation t Subscript b comma i Baseline plus sigma-summation Underscript b element-of upper M Endscripts k Subscript b Baseline dot b

of the map mu overTilde , draw an analytic arc gamma equals StartSet upper D Subscript lamda Baseline EndSet in the subvariety

normal upper Phi equals upper N overTilde plus sigma-summation s Subscript b comma i Baseline plus sigma-summation Underscript b element-of upper M Endscripts k Subscript b Baseline dot b plus sigma-summation left-parenthesis normal upper Delta Subscript b Baseline right-parenthesis Subscript k Sub Subscript b Baseline subset-of upper B Subscript 2 d plus 2 g prime minus 2

tending to the point

upper D 0 equals upper N overTilde plus sigma-summation s Subscript b comma i Baseline plus 2 sigma-summation Underscript b element-of upper M Endscripts k Subscript b Baseline dot b period

Since the image of the composition

ModifyingAbove upper M With bar Subscript g prime comma 0 Baseline left-parenthesis upper X comma beta right-parenthesis long right-arrow upper B Subscript 2 d plus 2 g prime minus 2

contains normal upper Phi , we can find an arc delta equals StartSet f Subscript nu Baseline EndSet in ModifyingAbove upper M With bar Subscript g prime comma 0 Baseline left-parenthesis upper X comma beta right-parenthesis that maps onto gamma , with f 1 the inclusion upper C overTilde right-arrow with hook upper X .

Stage 3: Let f 0 colon upper C 0 right-arrow upper X be the limit, in ModifyingAbove upper M With bar Subscript g prime comma 0 Baseline left-parenthesis upper X comma beta right-parenthesis , of the family of curves constructed in Stage 2; that is, the point of the arc delta over upper D 0 element-of normal upper Phi subset-of upper B Subscript 2 d plus 2 g prime minus 2 . Let upper A right-arrow upper C 0 be the normalization of any irreducible component of upper C 0 on which the composition pi f 0 is nonconstant (that is, whose image is not contained in a fiber), and let f Subscript upper A Baseline colon upper A right-arrow upper X be the stable map where f Subscript upper A is the restriction of f 0 to upper A . In particular, f Subscript upper A Baseline colon upper A right-arrow upper X is a stable map satisfying Hypothesis 2.7.

By construction, the composition mu Subscript upper A Baseline colon equals pi ring f Subscript upper A is unramified over a neighborhood of upper M : the monodromy around the boundary partial-differential normal upper Delta Subscript b of each disc normal upper Delta Subscript b is trivial, and it can be branched over at most one point b inside normal upper Delta Subscript b , so it cannot be branched at all over normal upper Delta Subscript b . Indeed, it is (at most) simply branched over each point of upper N and each point s Subscript b comma i , and unramified elsewhere. Moreover, since we can carry out the specialization of f overTilde colon upper C overTilde right-arrow upper X above with the entire fiber of mu overTilde over the points of upper N and the s Subscript b comma i fixed, the ramification of mu Subscript upper A over these points will occur at smooth points of the corresponding fibers of pi . In other words, the map f Subscript upper A Baseline colon upper A right-arrow upper X is pre-flexible, and we are done.

3.2. The proof for arbitrary curves upper B

As we indicated at the outset, there are two straightforward ways of extending this result to the case of arbitrary curves upper B .

For one thing, virtually all of the argument we have made goes over without change to the case of base curves upper B of any genus h . The one exception to this is the statement that the space ModifyingAbove upper M With bar Subscript g comma 0 Baseline left-parenthesis upper B comma d right-parenthesis of stable maps f colon upper C right-arrow upper B of degree d from curves upper C of genus g to upper B has a unique irreducible component whose general member corresponds to a flat map f colon upper C right-arrow upper B . This is false in general—consider for example the case g equals d left-parenthesis h minus 1 right-parenthesis plus 1 of unramified covers. It is true, however, if we restrict ourselves to the case g much-greater-than h comma d (that is, we have a large number of branch points) and look only at covers whose monodromy is the full symmetric group upper S Subscript d ReferenceGHS, Theorem 1. Given this fact and observing that our second construction allows us to increase the number of branch points of our covers upper C right-arrow upper B arbitrarily, the theorem can be proved for general upper B just as it is proved above for upper B approximately-equals double-struck upper P Superscript 1 .

Alternatively, Johan de Jong showed us a simple way to deduce the theorem for general upper B from the case upper B approximately-equals double-struck upper P Superscript 1 alone. We argue as follows: given a map pi colon upper X right-arrow upper B with rationally connected general fiber, we choose any map g colon upper B right-arrow double-struck upper P Superscript 1 expressing upper B as a branched cover of double-struck upper P Superscript 1 . We can then form the “norm” of upper X : this is the (birational isomorphism class of) variety upper Y right-arrow double-struck upper P Superscript 1 whose fiber over a general point p element-of double-struck upper P Superscript 1 is the product

upper Y Subscript p Baseline equals product Underscript q element-of g Superscript negative 1 Baseline left-parenthesis p right-parenthesis Endscripts upper X Subscript q Baseline period

Since the product of rationally connected varieties is again rationally connected, it follows from the double-struck upper P Superscript 1 case of the theorem that upper Y right-arrow double-struck upper P Superscript 1 has a rational section, and hence so does pi .

Acknowledgments

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.

Figures

Figure 1.

Specializing branch points

Mathematical Fragments

Theorem 1.1

Let f colon upper X right-arrow upper B be a proper morphism of complex varieties with upper B a smooth curve. If the general fiber of f is rationally connected, then f has a section.

Corollary 1.3

Let f colon upper X right-arrow upper Y be any dominant morphism of complex varieties. If upper Y and the general fiber of f are rationally connected, then upper X is rationally connected.

Corollary 1.4

Let upper X be any variety and phi colon upper X right-arrow upper Z its maximal rationally connected fibration. Then upper Z is not uniruled.

Conjecture 1.5

Let upper X be a smooth projective variety. Then upper X is uniruled if and only if upper H Superscript 0 Baseline left-parenthesis upper X comma upper K Subscript upper X Superscript m Baseline right-parenthesis equals 0 for all m greater-than 0 .

Conjecture 1.6

Let upper X be a smooth projective variety. Then upper X is rationally connected if and only if upper H Superscript 0 Baseline left-parenthesis upper X comma left-parenthesis normal upper Omega Subscript upper X Superscript 1 Baseline right-parenthesis Superscript circled-times m Baseline right-parenthesis equals 0 for all m greater-than 0 .

Hypothesis 2.1

pi colon upper X right-arrow upper B is a nonconstant morphism of smooth connected projective varieties over double-struck upper C , with upper B approximately-equals double-struck upper P Superscript 1 . For general b element-of upper B , the fiber upper X Subscript b Baseline equals pi Superscript negative 1 Baseline left-parenthesis b right-parenthesis is rationally connected.

Lemma 2.6

Let upper X be a smooth projective variety and let left-parenthesis f colon upper C right-arrow upper X comma left-parenthesis p 1 comma ellipsis comma p Subscript delta Baseline right-parenthesis right-parenthesis and let left-parenthesis f prime colon upper C right-arrow upper X comma left-parenthesis p prime 1 comma ellipsis comma p prime Subscript delta right-parenthesis right-parenthesis be as above. Let left-parenthesis upper F colon upper D right-arrow upper X right-parenthesis be the stable map obtained by gluing each p Subscript i to p prime Subscript i . The normal sheaf of the map script upper N Subscript upper F is locally free near each of the nodes p Subscript i of upper D and we have an inclusion of sheaves

0 right-arrow script upper N Subscript f Baseline right-arrow script upper N Subscript upper F Baseline vertical-bar Subscript upper C Baseline

identifying script upper N Subscript upper F Baseline vertical-bar Subscript upper C Baseline with the sheaf of rational sections of script upper N Subscript f having at most a simple pole at each p Subscript i in the normal direction determined by d f prime left-parenthesis upper T Subscript p prime Sub Subscript i Baseline upper C prime right-parenthesis . Moreover, if left-parenthesis upper F overTilde colon upper D overTilde right-arrow upper X right-parenthesis is a first-order deformation of upper F corresponding to a global section sigma element-of upper H Superscript 0 Baseline left-parenthesis upper D comma script upper N Subscript upper F Baseline right-parenthesis , then upper D overTilde smooths the node of upper D at p Subscript i if and only if the restriction sigma vertical-bar Subscript upper U Baseline of sigma to a neighborhood upper U of p Subscript i in upper C is not in the image of script upper N Subscript f .

Hypothesis 2.7

Suppose pi colon upper X right-arrow upper B is a morphism satisfying our basic Hypothesis 2.1. An unmarked stable map f colon upper C right-arrow upper X satisfies our second hypotheses if upper C is smooth and mu colon equals pi ring f is surjective.

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Article Information

MSC 2000
Primary: 14M20 (Rational and unirational varieties), 14D05 (Structure of families)
Author Information
Tom Graber
Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138
graber@math.harvard.edu
Joe Harris
Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138
harris@math.harvard.edu
Jason Starr
Department of Mathmatics, Massachusetts Institute of technology, Cambridge, Massachusetts 02139
jstarr@math.mit.edu
Additional Notes
The first author was partially supported by an NSF Postdoctoral Fellowship. The second author was partially supported by NSF grant DMS9900025. The third author was partially supported by a Sloan Dissertation Fellowship.
Journal Information
Journal of the American Mathematical Society, Volume 16, Issue 1, ISSN 0894-0347, published by the American Mathematical Society, Providence, Rhode Island.
Publication History
This article was received on , revised on , and published on .
Copyright Information
Copyright 2002 American Mathematical Society
Article References