Ample subvarieties and rationally connected fibrations
AMPLE SUBVARIETIES AND RATIONALLY CONNECTED
FIBRATIONS
MAURO C. BELTRAMETTI, TOMMASO DE FERNEX, AND ANTONIO LANTERI
Abstract. Under some positivity assumptions, extension properties of rationally con-
nected fibrations from a submanifold to its ambient variety are studied. Given a family
of rational curves on a complex projective manifold X inducing a covering family on a
submanifold Y with ample normal bundle in X, the main results relate, under suitable
conditions, the associated rational connected fiber structures on X and on Y . Applica-
tions of these results include an extension theorem for Mori contractions of fiber type
and a classification theorem in the case Y has a structure of projective bundle or quadric
fibration.
Introduction
Both from a biregular and a birational standpoint, the geometry of algebraic varieties
is often studied in terms of their fibrations. Given a smooth complex projective variety
X, there are two parallel theories able to detect on X fibrations whose fibers are varieties
of negative Kodaira dimension. The first one, initiated by Mori, associates a morphism
X → S, whose general positive dimensional fiber is a Fano variety, to any KX-negative
extremal ray of the cone of effective curves of X. The second theory, introduced in works
of Campana and of Kollár, Miyaoka and Mori, produces a rational map X 99K S, with
proper and rationally chain connected fibers within its domain of definition, from any
family of rational curves on X, the most studied case being the one in which the family
of all rational curves of X is considered.
The purpose of this paper is to study extension properties of fibrations of the types
described above, when these fibrations are defined on a smooth subvariety Y of X with
ample normal bundle: our goal is to determine additional conditions that guarantee that
given fiber structures on Y extend to analogous structures on X, and to compare the
corresponding fibrations. Following the terminology of [13], we will refer to such a Y as
an “ample subvariety” of X. We remark that, in the special case of codimension one, this
setting is more general than the classical setting of ample divisors.
Our first result is an extension property for rationally connected fibrations. The pre-
cise formulation requires some additional notation, and is given in Theorems 3.1 and 3.6.
1991 Mathematics Subject Classification. Primary: 14D06, 14J10. Secondary: 14C05, 14J40, 14N30.
Key words and phrases. Ample subvarieties, rationally connected fibrations, families of rational curves,
special varieties, extension of maps, Mori contractions.
All authors acknowledge support by MIUR National Research Project “Geometry on Algebraic Va-
rieties” (Cofin 2004). The research of the second author was partially supported by NSF grants DMS
0111298 and DMS 0548325. The third author acknowledges partial support by the University of Milan
(FIRST 2003).
http://arxiv.org/abs/0704.0661v3
2 M.C. BELTRAMETTI, T. DE FERNEX, AND A. LANTERI
Roughly speaking, Theorem 3.1 says that if Y is a submanifold with ample normal bundle,
the inclusion inducing a surjection N1(X) → N1(Y ), and V is a family of rational curves
on X inducing a covering family VY on Y , then there is a commutative diagram
Y//VY
// X//V ,
where φ is the rational map associated to the family V , π is the map associated to VY , and
δ is a surjective morphism of normal varieties; here X//V and Y//VY denote the respective
rational quotients.
With the second theorem, we determine sufficient conditions to ensure that the
morphism δ in the above diagram is generically finite; this is one of the core results of the
paper (see Theorem 3.6).
Theorem A. Assuming that the normal bundle of Y in X is ample and that the inclusion
of Y induces a surjection N1(X) ։ N1(Y ), the morphism δ is generically finite if one of
the following two conditions is satisfied:
(a) V is an unsplit family; or
(b) codimX Y < dimY − dimY//VY .
When the rational quotient Y//VY is one-dimensional, it turns out from a general
fact that π is a morphism (see Proposition 3.12); using this fact, the previous result can
be improved in this case, and we obtain a commutative diagram as the one above where
now the vertical arrows are morphisms and δ is a finite morphism between smooth curves
(see Corollary 3.13). The situation when Y is the zero scheme of a regular section of an
ample vector bundle on X and π is the MRC-fibrations over a base of positive geometric
genus was also treated in [25] (see Remark 3.10).
Next, we address the problem of extending extremal Mori contractions of fiber-type.
Using the previous result, we prove the following theorem (see Theorem 4.1).
Theorem B. Let Y be a submanifold of a projective manifold X with ample normal
bundle, and assume that the inclusion induces an isomorphism N1(X) ∼= N1(Y ). Let
π : Y → Z be an extremal Mori contraction of fiber-type, and let W be an irreducible
covering family of rational curves on Y that are contracted by π. If these curves do not
break in X under deformation (that is, if the family they generate in X is unsplit), then π
extends to an extremal Mori contraction φ : X → S, and there is a commutative diagram
// S,
where δ is a finite surjective morphism.
In the special case when Y is defined by a regular section of an ample vector bundle
on X, related results were obtained in [1, 9, 25] (see Remark 4.3).
AMPLE SUBVARIETIES AND RATIONALLY CONNECTED FIBRATIONS 3
In the last section of the paper, the above results are finally applied to classify, un-
der suitable conditions, projective manifolds containing “ample subvarieties” that admit
a structure of projective bundles or quadric fibrations. Beginning with classical results on
hyperplane sections, there has been given evidence to the fact that projective manifolds
are, so to speak, at least as “special” as their ample divisors (cf. [26]). The study of pro-
jective bundles and quadric fibrations embedded in projective manifolds as zero schemes
of regular sections of ample vector bundles was already undertaken in [18, 19, 20, 8, 25],
where classification results were obtained when the base of the fibration, if positive dimen-
sional, has positive geometric genus. Under the additional assumption of a polarization
on the ambient variety inducing a relatively linear polarization on the fibration, classifi-
cation results were obtained in [7, 1, 21]. With some dimensional restrictions on the fiber
structure of the subvariety in terms of its codimension, we extend such results in two ways.
In our first generalization, we consider fibrations over curves and drop the hypothesis
that the subvariety is defined by a regular section of an ample vector bundle, only assuming
that the normal bundle is ample. In this case, since the positivity condition is local near Y ,
we additionally require that the inclusion induces an isomorphism on the Picard groups. In
the second generalization, we allow the base of the projective fibration to have arbitrary
dimension. In either case, we do not put any additional restriction on the base of the
fibration and do not require a priori any global polarization (such a polarization will turn
out to exist a posteriori). We state here the extension property that we obtain in the case
of projective bundles.
Theorem C. Let Y be a submanifold of a projective manifold X with ample normal
bundle, of codimension
(1) codimX Y < dimY − dimZ.
Assume that Y admits a projective bundle structure π : Y → Z over a smooth projective
variety Z, and that one of the following two situations occurs:
(a) Z is a curve and the inclusion of Y in X induces an isomorphism Pic(X) ∼= Pic(Y );
(b) Y is the zero scheme of a regular section of an ample vector bundle on X.
Then π extends to a projective bundle structure π̃ : X → Z on X, giving a commutative
diagram
eπ~~~~
and the fibers of π are linearly embedded in the fibers of π̃.
A similar statement is also proven in the case of quadric fibrations with irreducible
reduced fibers and relative Picard number one. For the full result, see Theorem 5.8.
The proofs of the above results rely on various properties concerning families of
rational curves on projective manifolds and their associated rational quotients, which are
collected in Section 2. General facts from deformation theory of rational curves needed in
the sequel are recalled in this section, and several new notions are introduced and studied.
4 M.C. BELTRAMETTI, T. DE FERNEX, AND A. LANTERI
Among other things, we study numerical properties of families of rational curves, intro-
ducing in particular the notion of numerically covering family (see Definition 2.18), which
turns out to be very useful in our context. Appropriate notions for the extension and re-
striction of families of rational curves are also introduced in relation to given submanifolds
of the ambient variety.
Acknowledgements. We would like to thank C. Araujo, L. Bădescu, C. Casagrande and
P. Ionescu for useful discussions, and the referee for many precious remarks and sugges-
tions. We are grateful to the Istituto Nazionale di Alta Matematica and to the University
of Milan (FIRST 2003) for making this collaboration possible.
1. Conventions and basic notation
We work over the complex numbers, and use standard notation in algebraic geome-
try, although tensor product between line bundles is often denoted additively. For n ≥ 1,
we denote by Qn the smooth quadric hypersurface of Pn+1. We use the word general to
mean that the choice is made outside a properly contained (Zariski) closed subset, and
the word very general to mean that the choice is made outside a countable union of such
subsets. For any projective algebraic variety X, we denote by N1(X) := (Pic(X)/ ≡)⊗R
the Néron–Severi space of X, and by N1(X) := (Z1(X)/ ≡) ⊗ R the space generated by
numerical classes of curves on X. The dimension of these spaces, which is equal to the
Picard number of X, is denoted by ρ(X). For a morphism of algebraic varieties X → S,
we set N1(X/S) := (Z1(X/S)/ ≡) ⊗ R, and we denote by ρ(X/S) the dimension of this
space. The numerical class of a line bundle L (resp., of a curve C) on X is denoted by [L]
(resp., [C]). We denote by NE(X) the closure of the cone in N1(X) spanned by numer-
ical classes of effective curves, and by Nef(X) its dual cone, namely, the cone in N1(X)
spanned by classes of nef divisors. In the case X is a projective variety with Pic(X) ∼= Z,
we will denote by OX(1) the ample generator of this group.
2. Families of rational curves on varieties
Throughout this section we consider morphisms from the projective line P1 to a
smooth projective variety X; for convenience, we fix once for all two distinct points 0 and
∞ in P1. We will use notation as in [15], some of which we recall below. For general facts
on the theory of deformation of rational curves on varieties, we refer to [15, 6].
Let Hom(P1,X) be the scheme parameterizing morphisms from P1 to X. We will
denote a morphism f : P1 → X by [f ] whenever we think of it as a point of Hom(P1,X).
If Σ ⊆ P1 and Θ ⊆ X are closed subschemes, then we denote by
Hom(P1,X; Σ → Θ)
the subscheme of Hom(P1,X) parameterizing those morphisms f such that f(Σ) ⊆ Θ (the
image f(Σ) of Σ is here intended scheme-theoretically). In the particular case Σ = {0}
and Θ = {x} for some x ∈ X, we also use the notation Hom(P1,X; 0 7→ x). Note that,
for any closed subscheme j : Σ →֒ P1, there is a natural morphism
j∗ : Hom(P1,X) → Hom(Σ,X)
defined by j∗([f ]) := [f ◦ j].
AMPLE SUBVARIETIES AND RATIONALLY CONNECTED FIBRATIONS 5
Definition 2.1. A morphism f : P1 → X is free if it is non-constant and f∗TX is nef.
We will use the following well-known properties (see [15, Theorem II.1.7]).
Proposition 2.2. Let f : P1 → X be a non-constant morphism, and let x = f(0). Let
m0 ⊂ OP1 be the maximal ideal sheaf of 0 ∈ P
(a) The Zariski tangent space of Hom(P1,X) at [f ] is isomorphic to H0(P1, f∗TX),
and Hom(P1,X) is smooth at [f ] if h1(P1, f∗TX) = 0.
(b) Similarly, the Zariski tangent space of Hom(P1,X; 0 7→ x) at [f ] is isomorphic to
H0(P1, f∗TX⊗m0), and Hom(P
1,X; 0 7→ x) is smooth at [f ] if h1(P1, f∗TX⊗m0) =
In particular, if f is free, then Hom(P1,X) and Hom(P1,X; 0 7→ x) are smooth at [f ].
For closed subschemes Θ ⊆ X and Σ ⊆ P1, and any subset S ⊆ Hom(P1,X), we
denote
S(Σ → Θ) := S ∩Hom(P1,X; Σ → Θ),
and, in a similar way, we define S(0 7→ x) for any point x ∈ X.
Definition 2.3. Given a subset S ⊆ Hom(P1,X), the image of the universal map P1×S →
X is called locus of S, and is denoted by Locus(S). Analogous definitions are given for
the loci of S(Σ → Θ) and S(0 7→ x), which are respectively denoted by Locus(S; Σ → Θ)
and Locus(S; 0 7→ x).
We now restrict our attention to the open subscheme Hombir(P
1,X) of Hom(P1,X)
parameterizing morphisms that are birational to their images. The previous notation is
adapted to Hombir(P
1,X) in the obvious way.
Definition 2.4. Any union V = ∪α∈AVα of irreducible components Vα of Hombir(P
is said to be a family of (parameterized) rational curves on X. If V consists of only one
irreducible component of Hombir(P
1,X), then V is said to be an irreducible family. If
Locus(Vα) is dense in X for every α ∈ A, then V is said to be a covering family.
Remark 2.5. The scheme Hombir(P
1,X) has at most countably many irreducible compo-
nents. In particular, the same holds for every family of rational curves on X.
Proposition 2.6. Let V be an irreducible family of rational curves on X. Then the locus
Locus(V ) of V is dense in X if and only if f is free for a general [f ] ∈ V .
Proof. Locus(V ) is dense in X if and only if the differential of the universal map P1×V →
X has rank equal to dimX at a general point (p, [f ]) ∈ P1×V . By [15, Proposition II.3.10],
this occurs if and only if f is free for a general [f ] ∈ V . �
Definition 2.7. Given any closed subset S ⊆ Hombir(P
1,X), we denote by 〈S〉 the union
of all irreducible components of Hombir(P
1,X) that contain at least one irreducible com-
ponent of S, and by 〉S〈 the union of all irreducible components of Hombir(P
1,X) that
are irreducible components of S. We call 〈S〉 the minimal family generated by S, and 〉S〈
the maximal family contained in S.
Analogous notation and definitions will be adopted for S ⊆ Hom(P1,X).
6 M.C. BELTRAMETTI, T. DE FERNEX, AND A. LANTERI
Definition 2.8. Let V be a family of rational curves on X. We say that a curve C ⊂ X
is a V -curve if C = f(P1) for some [f ] ∈ V . A V -chain of length ℓ is the union of ℓ
distinct curves fi(P
1) ⊂ X (1 ≤ i ≤ ℓ) parameterized by elements [fi] ∈ V such that
fi+1(0) = fi(∞) for every 1 ≤ i ≤ ℓ− 1.
Consider a family V = ∪Vα of rational curves on X. Let V
α be the irreducible
components of the normalization Homnbir(P
1,X) of Hombir(P
1,X) that map to Vα, and let
V be the closure in Chow1(X) of the image of ∪V
α via the natural morphism
(2) Homnbir(P
1,X) → Chow1(X)
(see [15, Comment II.2.7]). By [15, Theorem I.5.10], V is proper. Associated to V (hence
to V ), one can define a proper proalgebraic relation on X, as explained in [15, Section IV.4]
(see in particular [15, Example IV.4.10]). We denote this relation by RCV . We have that
(x, y) ∈ RCV for two very general points x, y ∈ X if and only if there is a V -chain
containing x and y.
Definition 2.9. X is said to be RCV -connected if there is only one RCV -equivalence
class. We say that X is V -chain connected if every two points of X lie on a V -chain. More
generally, a closed subset T ⊂ X is said to be V -chain connected if any two points of T
lie on a V -chain supported on T .
Clearly if X is V -chain connected, then it is RCV -connected.
Definition 2.10. A family of rational curves V is said to be an unsplit family if it is
irreducible and, after normalization, its image in Chow1(X) via the map (2) is a proper
scheme.
Remark 2.11. If V is an irreducible family of rational curves on X, and there is an ample
vector bundle E onX such that detE·C < 2 rkE for a V -curve C, then V is unsplit. Indeed,
i=1[Ci] is any degeneration of the cycle [C], then detE · C =
i=1 detE · Ci ≥ k rkE
by the ampleness of E. Thus k = 1 and [C1] lies in the image of the normalization of V in
Chow1(X) via the map (2).
Proposition 2.12. If X is RCV -connected by a family V = ∪Vα with each Vα unsplit,
then it is V -chain connected and N1(X) is generated by numerical classes of V -curves. In
particular, if V is unsplit, then X has Picard number ρ(X) = 1.
Proof. Fix ℓ≫ 0 such that every two very general points of X are connected by a V -chain
of length ℓ, and let S ⊂ Chow1(X) be the subset parameterizing connected 1-cycles with
ℓ components (counting multiplicities) supported on V -curves. Since each component Vα
of V is unsplit, it follows that S is proper. By the choice of ℓ, there is an irreducible
component T of S such that u(2) : U ×T U → X × X is dominant, where U → T is the
universal family and u : U → X is the cycle map. Note that T is a closed subvariety of
Chow1(X), and in particular it is proper. Therefore the image of u
(2) is proper, and thus
u(2) is surjective. This implies that X is V -chain connected. The second assertion follows
from [15, Proposition IV.3.13.3]. �
If V = Hombir(P
1,X), then it is a result of Campana [5] and Kollár–Miyaoka–Mori
[16] that one can “pass to the quotient”. The natural generalization of this result when V is
AMPLE SUBVARIETIES AND RATIONALLY CONNECTED FIBRATIONS 7
an arbitrary family of rational curves was later given by Kollár (see [15, Theorem IV.4.17]),
and can be stated as follows.
Theorem 2.13. Let V be a family of rational curves on X. Then there is an open set
X◦ ⊆ X, a normal projective variety X//V , and a dominant morphism X
◦ → X//V with
connected fibers and proper over the image, whose very general fibers are RCV -equivalence
classes in X.
Proof. We refer the reader to [15, Theorem IV.4.17] for the existence of a proper morphism
X◦ → Y ◦ onto a variety Y ◦ with connected fibers and whose very general fibers are RCV -
equivalence classes in X. Since X◦ is normal and the fibers are connected, Y ◦ is normal.
It follows by construction that Y ◦ is quasi-projective. Then we can take X//V to be the
normalization of the closure of Y ◦ in some projective embedding. �
Definition 2.14. With the notation as in the previous theorem, the resulting rational
map X 99K X//V is called the RCV -fibration of X, and X//V is the RCV -quotient. We also
say that X//V (resp., X 99K X//V ) is the rational quotient (resp., the rational fibration)
defined by V . We denote by
dimRCV := dimX − dimX//V
the dimension of a very general RCV -equivalence class (which is the same as the dimension
of a general fiber of X 99K X//V ).
Remark 2.15. We have dimRCV > 0 if and only if V is a covering family. We would
like to stress that X//V is well defined only up to birational equivalence; throughout the
paper, we will often make suitable choices of rational quotients. We remark that, after
possibly shrinking further X◦, one can always take a smooth projective model for the
rational quotient X//V .
Remark 2.16. A case of particular interest is when V = Hombir(P
1,X). In this case X//V
is called the MRC-quotient (maximal rational quotient) of X. It follows from a result of
Graber, Harris and Starr [12] that in this situation X//V is not uniruled.
Every connected subset S ⊆ Hombir(P
1,X) determines in a natural way a numerical
class [S] ∈ N1(X), by taking the class of the curve f(P
1) for an arbitrary [f ] ∈ S. More
generally, we give the following definition.
Definition 2.17. For any subset S ⊆ Hombir(P
1,X), we denote
R≥0[S] :=
[f ]∈S
R≥0[f(P
1)] ⊂ N1(X).
We call R≥0[S] the cone numerically spanned by S. If S is connected, then we call [S] the
numerical class of S.
Definition 2.18. A family of rational curves V = ∪α∈AVα is said to be numerically
covering if there is a subset A′ ⊆ A such that
R≥0[V ] =
R≥0[Vα]
8 M.C. BELTRAMETTI, T. DE FERNEX, AND A. LANTERI
and Locus(Vα) is dense in X for every α ∈ A
′. If V is a numerically covering family, then
we denote by Vcov the subfamily of V consisting of all the covering irreducible components
of V .
Remark 2.19. Note that by definition R≥0[Vcov] = R≥0[V ] for every numerically covering
family V of rational curves on X. Clearly, in the notation of the definition, we have
∪α∈A′Vα ⊆ Vcov.
Proposition 2.20. Let V = ∪α∈AVα and W = ∪β∈BWβ be two numerically covering
families of rational curves on X, and suppose that
R≥0[V ] ⊆ R≥0[W ].
Then the RCW -fibration ψ : X 99K X//W factors through the RCV -quotient of X, i.e., we
have a commutative diagram
X//V //___ X//W .
In particular, if R≥0[V ] = R≥0[W ], then V and W define the same rational fibration.
Proof. Consider a resolution of the indeterminacies of ψ
ψ ""F
X//W ,
and let E ⊂ X ′ be the exceptional locus of σ. Note that the locus of indeterminacies of
ψ is contained in X \X◦, and hence, by Theorem 2.13, it does not dominate X//W , since
it does not meet the very general RCV -equivalence class. Therefore we can assume that
ψ′(E) is a proper closed subset of X//W .
Lemma 2.21. If C ⊂ X is a V -curve not contained in σ(E), then its proper transform
C ′ ⊂ X ′ is mapped to a point in X//W by ψ
Proof of the lemma. Let B′ ⊂ B be the subset parameterizing the irreducible components
of W that are covering, so that Wcov = ∪β∈B′Wβ. Since
R≥0[V ] ⊆ R≥0[W ] = R≥0[Wcov],
we can pick general elements [gβ] ∈ Wβ for β ∈ B
′, and numbers λβ ≥ 0, such that,
denoting Γβ = gβ(P
1), we have
[C] =
λβ[Γβ] in N1(X).
By the definition of ψ and Theorem 2.13 and the fact that the Wβ are covering families
for every β ∈ B′, we can assume that Γβ ∩ σ(E) = ∅ for every β. Let C
′ and Γ′β be the
proper transforms of C and Γβ on X
′, and let D ⊂ X ′ be the pull-back, via ψ′, of an
AMPLE SUBVARIETIES AND RATIONALLY CONNECTED FIBRATIONS 9
ample divisor on X//W . Then ψ
β ] = 0, so we have σ∗D ·Γβ = D ·Γ
β = 0 for all β, and
D · C ′ ≤ σ∗D · C =
λβ(σ∗D · Γβ) = 0.
Since D is the pull-back of an ample divisor on X//W , this implies that ψ
′] = 0. �
We can now conclude the proof of the proposition. We consider a very general fiber
G of φ, and fix two very general points x, y ∈ G that are not contained in σ(E). By
the generality of the choices, we can assume that both x and y are outside the locus of
indeterminacies of ψ, hence ψ(x), ψ(y) 6∈ ψ′(E). Furthermore, we can assume that x and
y are connected by a chain of V -curves. By Lemma 2.21, every irreducible component
of this chain that is not contained in σ(E) is mapped to a point by ψ. Therefore, since
ψ(x) 6∈ ψ′(E), we deduce that this chain is in fact disjoint from σ(E). Thus we can apply
the lemma to each one of its components. Since the chain is connected, this implies
ψ(x) = ψ(y).
In view of the generality of the choice of x and y in G, we conclude that there exists a
natural rational map X//V 99K X//W commuting with the respective projections. �
Definition 2.22. Let V ⊆ Hombir(P
1,X) be any family of rational curves, and let a > 0
be an integer. We denote by ‖V ‖ the largest family of rational curves on X such that
R≥0[‖V ‖] = R≥0[V ]. We call ‖V ‖ the family numerically generated by V .
Remark 2.23. Quite obviously, V is a subfamily of ‖V ‖, and if V is a numerically covering
family then so is ‖V ‖.
The previous proposition implies the following useful property.
Corollary 2.24. Let V be a numerically covering family of rational curves on X. Then
the families V , Vcov, and ‖V ‖ define the same rational fibration.
Proof. Since R≥0[‖V ‖] = R≥0[V ] = R≥0[Vcov], it follows from Proposition 2.20. �
An important theorem, due to Kollár, Miyaoka and Mori, says that a smooth pro-
jective manifold is maximally rationally chain connected if and only if it is maximally
rationally connected (see [15, Theorem IV.3.10]). The proof of this property can be
adapted to the situation in which an arbitrary family of rational curves V is considered.
More precisely, one can prove that if X is V -chain connected, then every two very general
points of X are connected by a ‖V ‖-curve. In the sequel, we will need the following slightly
different version of this property (the proofs of the two properties are almost the same).
Proposition 2.25. Let V be a family of rational curves on X, and assume that X is
V -chain connected. Let y ∈ X be any point that is connected by a V -chain to a very
general point of X. Then Locus(‖V ‖, 0 7→ y) is dense in X.
Proof. Fix a very general point x ∈ X, and let C = C1+ · · ·+Cn be a V -chain connecting
x to y, with Ci = fi(P
1). Let p0 = x, pn = y, and pi = fi(∞) = fi+1(0) for 1 ≤
i ≤ n − 1. From here, we follow step by step the proof of [15, Complement IV.3.10.1],
proving inductively that there is a free rational curve gi : P
1 → X which connects pi−1
to pi; the resulting chain of free rational curves is then smoothed into a free rational
10 M.C. BELTRAMETTI, T. DE FERNEX, AND A. LANTERI
curve h : P1 → X connecting x to y. The construction shows that each gi+1 is obtained
by smoothing a comb on fi+1 with teeth assembled out of deformations of gi (see [15,
Definition II.7.7]). It follows that h is a ‖V ‖-curve. �
Consider now a submanifold Y of X. The inclusion i : Y →֒ X naturally induces an
injective morphism
i∗ : Hom(P
1, Y ) →֒ Hom(P1,X),
which is defined by i∗([g]) := [i ◦ g]. Similar notation will be used for the injection
Hombir(P
1, Y ) →֒ Hombir(P
1,X). Statements analogous to the following two propositions
also hold for Hombir( ) in place of Hom( ).
Proposition 2.26. With the above notation, assume that the normal bundle of Y in X
is ample. Then, for every [g] ∈ Hom(P1, Y ) with g free, the morphism f := i ◦ g is free on
X, and the schemes Hom(P1,X) and Hom(P1,X; 0 7→ g(0)) are smooth at [f ].
Proof. The bundle g∗TY is nef, since [g] is free. Thus, by the ampleness of NY/X and the
exact sequence
0 → g∗TY → f
∗TX → g
∗NY/X → 0,
we conclude that f is free. Then the last two assertions follow from Proposition 2.2. �
We will need the following generalization of Proposition 2.2.
Proposition 2.27. Let Y be a submanifold of a smooth projective variety X and let
i : Y →֒ X be the inclusion. Let g : P1 → Y be a free morphism, and let f := i ◦ g.
Then the Zariski tangent space of Hom(P1,X; {0} → Y ) at [f ] sits naturally in an exact
sequence
0 → H0(P1, g∗TY ) → T[f ]Hom(P
1,X; {0} → Y ) → H0(P1, g∗NY/X ⊗m0).
Moreover, if the normal bundle of Y in X is ample, then the sequence completes to a short
exact sequence
0 → H0(P1, g∗TY ) → T[f ]Hom(P
1,X; {0} → Y ) → H0(P1, g∗NY/X ⊗m0) → 0
and Hom(P1,X; {0} → Y ) is smooth at [f ].
Proof. The natural maps f∗TX → g
∗NY/X → g
∗NY/X |{0}, passing to cohomology and
taking into account the isomorphism T[f ]Hom(P
1,X) ∼= H0(P1, f∗TX) given by Proposi-
tion 2.2, yield a natural homomorphism
r : T[f ]Hom(P
1,X) → (g∗NY/X)|{0}.
We have a Cartesian square
Hom(P1,X; {0} → Y )
// Hom(P1,X)
Hom({0}, Y ) // Hom({0},X).
Note that there are natural identifications Hom({0}, Y ) = Y and Hom({0},X) = X.
The Zariski tangent space of Hom(P1,X; {0} → Y ) at [f ], viewed as a subspace of
AMPLE SUBVARIETIES AND RATIONALLY CONNECTED FIBRATIONS 11
T[f ]Hom(P
1,X), is contained in ker(r). This simply follows from the fact that the mor-
phism Hom(P1,X; {0} → Y ) → Hom({0},X) factors through Hom({0}, Y ), by taking
tangent spaces and recalling the above natural identifications. Now, the freeness of g
implies that H1(P1, g∗TY ) = 0, hence we get the exact sequence
0 → H0(P1, g∗TY ) → T[f ]Hom(P
1,X) → H0(P1, g∗NY/X) → 0.
This sequence restricts to an exact sequence
(3) 0 → H0(P1, g∗TY ) → ker(r) → H
0(P1, g∗NY/X ⊗m0) → 0,
and the first assertion follows by observing that T[f ]Hom(P
1,X; {0} → Y ), viewed as a
subspace of ker(r), contains the image of H0(P1, g∗TY ).
Suppose now that NY/X is ample. By Proposition 2.26, Hom(P
1,X; 0 7→ g(0)) is
smooth, hence of dimension h0(P1, f∗TX⊗m0), at [f ]. Let V be the irreducible component
of Hom(P1,X) containing [f ]. Note that T[f ]Hom(P
1,X; {0} → Y ) = T[f ]V ({0} → Y ).
Moreover, there is a dominant morphism V ({0} → Y ) → Y , defined by [h] 7→ h(0), whose
fiber over a point y ∈ Y is V (0 7→ y). This implies that dimV ({0} → Y ) = dimV (0 7→
y) + dimY for a general y ∈ Y , and therefore
(4) dimV ({0} → Y ) = h0(P1, f∗TX ⊗m0) + dimY
by Proposition 2.2,(b). On the other hand, we have
h0(P1, g∗NY/X ⊗m0) = h
0(P1, f∗TX ⊗m0)− h
0(P1, g∗TY ⊗m0)
= h0(P1, f∗TX ⊗m0)− h
0(P1, g∗TY ) + h
0(g∗TY |{0}).
Therefore, by (3) and h0(g∗TY |{0}) = dimY , we get
(5) dimker(r) = h0(P1, f∗TX ⊗m0) + dimY.
Then, comparing (5) with (4), we conclude at once that T[f ]Hom(P
1,X; {0} → Y ) = ker(r)
and that V ({0} → Y ) is smooth at [f ]. �
Definition 2.28. If W ⊆ Hombir(P
1, Y ) is a family of rational curves on Y , then we
call 〈i∗(W )〉 ⊆ Hombir(P
1,X) the extension of W to X. Conversely, for every family of
rational curves V on X, we call 〉i−1∗ (V )〈 ⊆ Hombir(P
1, Y ) the restriction of V to Y .
Remark 2.29. If V is an irreducible family on X, then 〉i−1∗ (V )〈 needs not be irreducible.
In fact, the example of the restriction to a smooth quadric Q2 of the family of lines
in P3 shows that, in general, different elements in 〉i−1∗ (V )〈 may even define linearly
independent numerical classes in N1(Y ). In particular, if V is an unsplit family on X,
then its restriction to Y may not be an unsplit family. Although we do not have examples,
it seems likely that i−1∗ (V ) might fail to be a family on Y even if V is a family on X.
We close this section with the following “relative” version of Proposition 2.12. The
proof, which can be found within the proof of [4, Lemma 1.4.5], uses a “non-breaking
lemma” due to Wísniewski [28].
Proposition 2.30. Let V be an unsplit family on X. Let Y ⊂ X be a subvariety, and
assume that Locus(V, {0} → Y ) is dense in X. Then, for every curve Γ ⊂ X, we have
[Γ] = a[ΓY ] + b[C] in N1(X),
12 M.C. BELTRAMETTI, T. DE FERNEX, AND A. LANTERI
where C is a V -curve, ΓY is a curve contained in Y , and a ≥ 0.
3. Extension of rationally connected fibrations
In this section we study the relationship between rational connected fibrations of a
projective manifold X and those of an ample submanifold Y ⊂ X. Our main interest is in
the case when it is given on X an irreducible family of rational curves V whose restriction
VY to Y contains a covering component. Given this situation, the goal is to compare
the associated rationally connected fibrations X 99K X//V and Y 99K Y//VY . However,
for technical reasons that will be evident in the proof of Theorem 3.6, it is convenient
to consider from the beginning a more general setting, allowing V to be reducible. The
conditions given in items (i) and (ii) in the following two theorems capture the essential
properties of V needed in our arguments.
Theorem 3.1. Let X be a smooth projective variety, and assume that Y ⊂ X is a smooth
subvariety with ample normal bundle. Let i : Y →֒ X be the inclusion, and suppose that the
induced map i∗ : N1(X) → N1(Y ) is surjective. Let V = ∪α∈AVα be a family of rational
curves on X, and assume that there is a subset B ⊆ A such that
(i) R≥0[V ] =
β∈B R≥0[Vβ ], and
(ii) Locus
i−1∗ (Vβ)
is dense in Y for every β ∈ B.
Let VY := 〉i
∗ (V )〈 be the restriction of V to Y . Then both V and VY are numerically
covering families (respectively, on X and on Y ), and, for suitable choices of the rational
quotients, there is a commutative diagram
Y//VY
// X//V ,
where π and φ are the projections to the respective rational quotients and δ is a surjective
morphism.
Remark 3.2. If one assumes that V is irreducible, then the hypothesis that i∗ : N1(X) →
N1(Y ) be surjective is unnecessary.
Remark 3.3. An analogous property has been independently observed to hold in the case
V = Hombir(P
1,X) by A.J. de Jong and J. Starr.
The proof of Theorem 3.1 is based upon the following two lemmas.
Lemma 3.4. With the same notation and assumptions as in Theorem 3.1,
Locus(Vβ; {0} → Y ) is dense in X for every β ∈ B.
Proof. To keep the notation light, we suppose throughout the proof that V is irreducible,
so that Vβ = V . By hypothesis, Locus(VY ) is dense in Y . Fix a general element [g] of
an irreducible component of VY with dense locus in Y , and let f := i ◦ g. Note that
[f ] ∈ V ({0} → Y ). Since the chosen component of VY has dense locus in Y , we can
assume that y := f(0) is a general point of Y . We know that g is free by Proposition 2.6.
AMPLE SUBVARIETIES AND RATIONALLY CONNECTED FIBRATIONS 13
Thus, since V is an irreducible component of Hom(P1,X) containing [f ], Proposition 2.27
applies to say that V ({0} → Y ) is smooth at [f ], with Zariski tangent space
T[f ]V ({0} → Y )
0(P1, g∗TY )⊕H
0(P1, g∗NY/X ⊗m0)).
We can view the right hand side as a vector subspace Λ ⊆ H0(P1, f∗TX) via the iso-
morphism (3) given in the proof of Proposition 2.27. It follows from the ampleness of
g∗NY/X that f
∗TX is spanned by the sections in Λ at every point q ∈ P
1 \ {0}. By [15,
Proposition I.2.19], we conclude that the differential of the universal map
1 × V ({0} → Y ) → X
has rank equal to dimX at (q, [f ]) for every q ∈ P1 \ {0}. Since V ({0} → Y ) is smooth at
[f ], this implies that its locus in X is dense. �
Lemma 3.5. With the same notation and assumptions as in Theorem 3.1, for every
β ∈ B the family Vβ is covering (in X) and every irreducible component of i
∗ (Vβ) with
dense locus in Y is a family (on Y ).
Proof. By the hypothesis (ii) of Theorem 3.1, we can fix an arbitrary irreducible component
T of i−1∗ (Vβ) with dense locus in Y . Let [g] ∈ T be a general element. We know by
Proposition 2.6 that g is free. Then by Proposition 2.26, f := i ◦ g is free and Hom(P1,X)
is smooth at [f ], and thus, in particular, Vβ is a covering family by Proposition 2.6 again.
Moreover, we deduce by the smoothness of Hom(P1,X) at [f ] that for every irreducible
one-parameter family [gt] in Hombir(P
1, Y ) specializing to [g], the corresponding family
[ft] := [i ◦ gt] is contained in V . Therefore [gt] ∈ i
∗ (V ), and in fact [gt] ∈ T by the
generality of the choice of [g] in the component T . We conclude that T is an irreducible
component of Hombir(P
1, Y ), and hence, in particular, of VY . �
Proof of Theorem 3.1. It follows from Lemma 3.5 and the hypothesis (i) that V is a nu-
merically covering family onX and similarly, using the fact that the i∗ : N1(Y ) → N1(X) is
injective, it follows that VY is a numerically covering family on Y . Let φ : X 99K Z//V and
π : Y 99K Y//VY be the rational quotients, as in Theorem 2.13. Let X
◦ ⊆ X, S◦ ⊆ X//V ,
Y ◦ ⊆ Y , and Z◦ ⊆ Y//VY be open subsets such that the maps φ and π restrict to proper
surjective morphisms
φ◦ : X◦ → S◦ and π◦ : Y ◦ → Z◦.
Let G be a very general fiber of φ◦, and let x be a general point of G. By Lemma 3.4, we
can assume that x ∈ Locus(V ; {0} → Y ), and hence we can find a point y ∈ Y such that
x ∈ Locus(V ; 0 7→ y) by Theorem 2.13. In particular, x and y are V -chain connected. This
implies that y ∈ G, and therefore that G ∩ Y 6= ∅. Since G is a general fiber of φ◦, this
means that φ◦(Y ∩X◦) = S◦. Moreover, by the generality of G, we can in fact assume that
G has non-empty intersection with a very general fiber of π◦. On the other hand, recalling
that π is defined by the restriction VY of the family V defining φ, we see that if F is a very
general fiber of π◦ meeting G, then necessarily F ⊆ G. In conclusion, after possibly further
14 M.C. BELTRAMETTI, T. DE FERNEX, AND A. LANTERI
shrinking Z◦ (and consequently Y ◦), we may assume to have a commutative diagram
// X◦
// S◦,
where δ◦ is a dominant morphism.
We need to show that, after possibly changing the birational model for Y//VY , the
map δ◦ extends to a surjective morphism
δ : Y//VY → X//V .
We first take the normalization Z of a projective compactification of Z◦. Note that δ◦
extends to a rational map Z 99K X//V , which is defined by some linear system H on Z.
Then we take Z ′ to be the normalization of the blow-up of the base scheme of H. Since the
proper transform of H on Z ′ is base-point free, the rational map δ◦ lifts to a well defined
morphism δ : Z ′ → X//V . This is a morphism of projective varieties, and δ
◦ is dominant,
so δ is surjective. Moreover, the morphism Z ′ → Z is an isomorphism over Z◦, and thus
we can identify the latter with a subset of Z ′. Then we take Z ′ as the VY -quotient Y//VY
of Y . This proves the theorem. �
Theorem 3.1 can be viewed as a general property of rationally connected fibrations.
Aiming also at applications in extension problems of specific fibrations, that will be ad-
dressed in the following sections, we are interested in determining sufficient conditions to
ensure that the map δ, whose existence was proven in the previous theorem, is generically
finite. This is the content of the next theorem, which is the main result of this section.
Theorem 3.6. Let X be a smooth projective variety, and assume that Y ⊂ X is a smooth
subvariety with ample normal bundle. Let i : Y →֒ X be the inclusion, and suppose that the
induced map i∗ : N1(X) → N1(Y ) is surjective. Let V = ∪α∈AVα be a family of rational
curves on X, and assume that there is a subset B ⊆ A such that
(i) R≥0[V ] =
β∈B R≥0[Vβ ], and
(ii) Locus
i−1∗ (Vβ)
is dense in Y for every β ∈ B.
Let VY := 〉i
∗ (V )〈 be the restriction of V to Y , and let
Y//VY
// X//V ,
be the commutative diagram given by Theorem 3.1. Suppose that one of the following two
conditions holds:
(a) V is an unsplit family; or
(b) codimX Y < dimRCVY .
Then the morphism δ is generically finite.
AMPLE SUBVARIETIES AND RATIONALLY CONNECTED FIBRATIONS 15
Proof. We first prove that δ is generically finite when the condition (a) of the statement is
satisfied. If Y//VY is a point, then the statement is obvious, so we can suppose that Y//VY
is positive dimensional. We suppose by contradiction that δ is not generically finite.
Keeping the notation introduced in the proof of Theorem 3.1, let G be a very
general fiber of φ◦, and consider YG := Y ∩ G. Note that G is smooth and rationally
connected by the restriction VG of V to G in the sense of Definition 2.28. So, since we
are assuming that V is unsplit, every irreducible component of VG is an unsplit family on
G, and hence N1(G) is generated by classes of VG-curves by Proposition 2.12. Since such
curves are all numerically equivalent in X, it follows that the image of the restriction map
Pic(X) → Pic(G) has rank 1. Consider the inclusions jX : G →֒ X, jY : YG →֒ Y and
iG : YG →֒ G. In order to reach a contradiction, we consider the commutative diagram
Pic(X)
// Pic(G)
Pic(Y )
// Pic(YG).
The plan is to give a lower-bound for the rank of the map Pic(X) → Pic(YG) by factoring
it through Pic(Y ).
Note that Y ◦G := Y
◦ ∩G is a non-empty open subset of YG. Since we are supposing
that δ◦ is not generically finite, we have dim
π◦(Y ◦G)
≥ 1. Therefore we can find a curve
Γ ⊆ YG such that Γ ∩ Y
G 6= ∅ and dim
π◦(Γ ∩ Y ◦G)
≥ 1. Since Locus(VY ) is dense in Y ,
every fiber of π◦ has positive dimension and, since these fibers are proper, we can fix a
curve C ⊂ YG lying inside a fiber of π
◦|Y ◦
Now, fix a projective model Y//VY for the rational quotient of Y containing Z
◦ as an
open subset, and let L ⊂ Y//VY be a general hyperplane section passing through a point
in π◦(Γ ∩ Y ◦G) but not containing the point π
◦(C). Let L◦ := L|Z◦ be the restriction of
L to Z◦, and let D ⊂ Y be the closure (in Y ) of (π◦)−1(L◦). If U ⊂ Z◦ \ L◦ is an open
neighborhood of π◦(C), then (π◦)−1(U) is an open neighborhood of C in X◦ (hence in
X) and is disjoint from (π◦)−1(L◦); this shows that D ∩ C = ∅. Let then DG and HG be
the restrictions of D and of a general hyperplane section H of Y to YG. By construction,
we have DG · Γ > 0 and DG · C = 0, whereas HG is ample; in particular, DG and HG
induce numerically independent elements of Pic(YG). Since both of them are restrictions
of divisors on Y , this implies that rk Im(j∗Y ) ≥ 2. Therefore, observing that the cokernel of
i∗ : Pic(X) → Pic(Y ) is torsion due to the surjectivity of N1(X) → N1(Y ), we conclude
rk Im(j∗Y ◦ i
∗) ≥ 2.
On the other hand, we have rk Im(i∗G ◦ j
X) ≤ 1, and therefore we have a contradiction
by the commutativity of the above diagram. This proves that δ is generically finite if the
condition (a) of the statement of the theorem is satisfied.
It remains to prove that δ is generically finite under the hypothesis (b). Let G be
a very general (smooth and connected) fiber of φ◦, and let F be a very general fiber of π
among those that are contained in G. Choosing G sufficiently general, we can also ensure
that F is general among the fibers of π◦. Then, to conclude the proof of the theorem, we
16 M.C. BELTRAMETTI, T. DE FERNEX, AND A. LANTERI
need to show that
(6) dimG = dimF + codimX Y.
Note indeed that, since dimG = dimRCV and dimF = dimRCVY , (6) implies that δ is
generically finite.
In order to prove (6), we fix a very general point y ∈ F . By Lemma 3.4,
Locus(V, {0} 7→ Y ) is dense in X. Moreover, we know that Locus(V, 0 7→ y) is con-
tained in G by Theorem 2.13. Thus Locus(V, 0 7→ y) sweeps a dense subset of G, if we let
y vary in F . Therefore, by our choices of F and y, we can assume that Locus(V, 0 7→ y)
contains a very general point of G. Note that any point of G is connected by a VG-chain
to the point y, where VG denotes the restriction of V to G in the sense of Definition 2.28.
Therefore Proposition 2.25 applied to G and VG implies that Locus
‖VG‖, 0 7→ y
is dense
in G. Note that the two families ‖V ‖(0 7→ y) and ‖VG‖(0 7→ y) are the same since G is
an RC‖V ‖ equivalence class. In particular, we have
dimG = dimLocus
‖V ‖, 0 7→ y
At this point the idea is to replace V with ‖V ‖, so that, by the above argument,
we can directly assume, without loss of generality, that the two sets Locus(V, 0 7→ y) and
G have the same dimension (and, in fact, that the first set is dense in the second one).
In order to make this step, we need to show, first of all, that ‖V ‖ satisfies hypotheses
analogous to (i) and (ii) imposed to V in the statement of the theorem, and moreover
that replacing V with ‖V ‖ does not affect the quotient maps φ and π and the respective
rational quotients, which implies condition (b) for ‖V ‖.
Lemma 3.7. The family ‖V ‖ satisfies the hypotheses (i) and (ii) imposed to V in Theo-
rem 3.6. Moreover, the families VY = 〉i
∗ (V )〈 and ‖V ‖Y := 〉i
∗ (‖V ‖)〈 define the same
rational fibration on Y . In particular, codimX Y < dimRC‖V ‖Y .
Proof of the lemma. First recall that VY is a numerically covering family by Theorem 3.1,
hence so is ‖VY ‖. We compare ‖V ‖ with
W := 〈i∗(‖VY ‖cov)〉.
Note that ‖VY ‖cov is non-empty and W is a subfamily of ‖V ‖. We claim that
(7) R≥0[‖V ‖] = R≥0[W ] in N1(X).
By the definition of ‖V ‖, this is equivalent to R≥0[V ] = R≥0[W ]. The inclusion R≥0[V ] ⊇
R≥0[W ] is obvious, so we need to show that the reverse inclusion holds. In fact, by the
hypothesis (i) on V , it suffices to show that R≥0[Vβ ] ⊆ R≥0[W ] for every β ∈ B. Since
Locus(i−1∗ (Vβ)) is dense in Y , at least one of the irreducible components of 〉i
∗ (Vβ)〈 is a
covering family on Y , hence a subfamily of ‖VY ‖cov. This implies that R≥0[Vβ] ⊆ R≥0[W ],
hence (7) is proven.
On the other hand, the restriction to Y of any irreducible component ofW is a union
of irreducible components of ‖VY ‖cov, which are covering by Definition 2.18. Therefore
their loci are dense in Y . Combining this with (7) proves the first assertion.
We are currently assuming that V satisfies the condition (b) in Theorem 3.6.
Therefore we get the desired inequality codimX Y < dimRC‖V ‖Y as soon as we show
AMPLE SUBVARIETIES AND RATIONALLY CONNECTED FIBRATIONS 17
that the families VY and ‖V ‖Y define the same rational quotient of Y , because then
dimRCVY = dimRC‖V ‖Y . By using Proposition 2.20 it is enough to show that
(8) R≥0[‖V ‖Y ] = R≥0[VY ] in N1(Y ).
The inclusion R≥0[‖V ‖Y ] ⊇ R≥0[VY ] is obvious. To prove the other, note that the embed-
ding of Y in X induces an inclusion
ι : N1(Y ) →֒ N1(X),
which follows from the hypothesis that i∗ : N1(X) → N1(Y ) is surjective made in Theo-
rem 3.6. Observe that
R≥0[‖V ‖Y ]
⊆ R≥0[‖V ‖] = R≥0[V ] =
R≥0[Vβ] = ι
R≥0[VY ]
in N1(X).
Since ι is injective, the inclusion holds inN1(Y ), before taking images. This proves equality
(8) and completes the proof of the lemma. �
We now come back to the proof of the theorem. By the previous lemma, we are
allowed to replace V with ‖V ‖. Therefore we can directly assume without loss of generality
(9) dimLocus(V, 0 7→ y) = dimG.
Then, in order to show (6) and hence to conclude the proof, it suffices to show that
dimLocus(V, 0 7→ y) = dimX − dimY//VY .
For short, let d = dimX and k = dimY//VY . Since y is a general point of Y , we can
assume that VY (0 7→ y) contains a sufficiently general point [g] of a covering component
of VY , so that g is free and C := g(P
1) is a smooth curve (the smoothness of C is not
essential for the proof, but simplifies the notation).
We observe that NY/X |F is an ample vector bundle over F , and that rkNY/X |F <
dimF by condition (b), and therefore H1((NY/X |F )
∗) = 0 by the Le Potier vanishing
theorem [23]. This implies that the short exact sequence
0 → NF/Y → NF/X → NY/X |F → 0
splits since NF/Y
F , and therefore we get a surjection NF/X ։ NF/Y . Restricting to
C and composing with the natural surjections TX |C ։ NC/X and NC/X ։ NF/X |C , we
obtain a chain of surjections
TX |C ։ NC/X ։ NF/X |C ։ NF/Y |C
Since g is free, so is f := i ◦ g by Proposition 2.26, and therefore TX |C is nef. Writing
TX |C ∼= ⊕iOP1(bi), we have bi ≥ 0, and thus the existence of a quotient of TX |C isomorphic
to O⊕k
implies that
(10) #{bi | bi = 0} ≥ k.
Note that (V, 0 7→ y) is smooth at [f ] by Proposition 2.2. Let u : P1×(V, 0 7→ y) → X
be the universal map, and let q ∈ P1 be a point different from 0. By [15, Proposition II.3.10]
and (10), we have
rkdu(q, [f ]) ≤ d− k.
18 M.C. BELTRAMETTI, T. DE FERNEX, AND A. LANTERI
Since (V, 0 7→ y) is smooth at [f ], we obtain dimLocus(V, 0 7→ y) ≤ d − k. On the
other hand, we have dimG ≥ d − k by Theorem 3.1. Thus by (9) we conclude that
dimLocus(V, 0 7→ y) = d− k, which proves (6). This concludes the proof of the theorem.
Remark 3.8. It is interesting to compare the result of Theorem 3.6 when the condition
in case (b) is satisfied with a result of Sommese [26, Proposition III] (see also [3, Theo-
rem (5.3.1)] for a more general version) in which the case of an ample divisor endowed with
a fibration is considered. Indeed, one notices that if codimX Y = 1, then the inequality
in (b) reduces exactly to the hypothesis imposed in the theorem of Sommese.
Remark 3.9. Hartshorne’s conjecture [13, Conjecture 4.5] on the intersection of ample
submanifolds with complementary dimensions, if true, would imply that the morphism δ
in Theorem 3.6 is in fact an isomorphism whenever the condition given in (b) is satisfied.
Indeed, let G be a general fiber of φ. Then Y ∩ G is a disjoint union of deg δ fibers Fj
of π, and by using standard exact sequences one easily sees that NY ∩G/G
∼= NY/X |Y ∩G.
In particular, we have NFj/G
∼= NY ∩G/G|Fj
∼= NY/X |Fj , which is ample. Note that con-
dition (b) implies that 2 dimFj > dimG. Then, assuming that deg δ ≥ 2, Hartshorne’s
conjecture would give the contradiction F1 ∩ F2 6= ∅; therefore δ can only be birational.
In fact, modulo suitable choices of the rational quotients, one could even assume that δ
is an isomorphism. We recall that this conjecture of Hartshorne is known to be true, for
instance, for homogeneous spaces [24]; we will use this fact in the proof of Theorem 5.8.
Remark 3.10. The previous results are related to a result of Occhetta [25, Proposition 4],
where an analogous extension property is obtained under more restrictive hypotheses. In
the setting of [25], Y is supposed to be the zero locus of a regular section of an ample vector
bundle on X, π is the MRC-fibration of Y , and the rational quotient of Y is assumed to
have positive geometric genus.
In applications, it might be useful to start with families of rational curves on Y ,
rather than only considering restrictions of families on X. The following elementary
property will be useful. Let X and Y be as above, with the inclusion inducing a surjection
N1(X) → N1(Y ), and consider a numerically covering family W of rational curves on
Y . Let then V = 〈i∗(W )〉 be the family on X obtained as the extension of W , and let
VY =〉i
∗ (V )〈 be its restriction back to Y . Note that VY is numerically covering since W
is so. We clearly have W ⊆ VY , but in general the inclusion may be strict (one can take
as an example the case of a quadric Y = Q2 in X = P3, taking W to be the family of
lines on Y corresponding to one of the two rulings). By this inclusion and Theorem 3.1,
we have a commutative diagram
// Y//VY
// X//V ,
for a suitable choice of the rational quotients, and some surjective morphisms γ and δ.
AMPLE SUBVARIETIES AND RATIONALLY CONNECTED FIBRATIONS 19
Proposition 3.11. Keeping the notation introduced above, both families W and VY define
the same rational quotient (i.e., Y//W = Y//VY and τ = π) if either one of the following
two situations occurs:
(a) dimY//W = dimX//V , or
(b) i∗ : N1(X) → N1(Y ) is surjective.
Proof. We observe that the general fiber of γ is rationally connected, since it is dominated
by the general fiber of π, and in particular it is connected. Then the conclusion in case (a)
follows immediately from the commutativity of the above diagram. We now focus on
case (b), and thus assume that N1(X) → N1(Y ) is surjective. By duality, we have an
injection N1(Y ) →֒ N1(X) and, by the construction of VY , we deduce that R≥0[VY ] =
R≥0[W ]. Then the assertion follows from Proposition 2.20. �
The case when the rational quotient is one-dimensional is particularly well suited
because of the following well-known property. We are grateful to the referee for suggesting
this proof, which simplifies our original argument.
Proposition 3.12. Let X be a normal projective variety, let φ◦ : X◦ → C be a dominant
morphism from a non-empty open subset X◦ ⊆ X to a smooth projective curve C, and
assume that φ◦ is proper over its image. Then φ◦ extends to a (surjective) morphism
φ : X → C.
Proof. We consider a projection C → P1. Composing with φ◦, we have a rational map
X 99K P1 which is defined by some linear pencil. Because of the properness of φ◦, this
pencil is free, hence the map X → P1 is regular and we can lift it via Stein factorization
to a regular morphism X → C which extends φ◦. �
Coming back to the extension of rationally connected fibrations, we immediately
obtain the following result.
Corollary 3.13. With the same notation and assumptions as in Theorem 3.6, suppose
furthermore that dimY//VY = 1. Then the RCVY -fibration π of Y and the RCV -fibration
φ of X are surjective morphisms fitting in a commutative diagram
Y//VY
// X//V ,
where δ is a finite morphism of smooth curves.
Proof. It follows from Theorems 3.1 and 3.6, and Proposition 3.12. �
4. Extension of Mori contractions
In this section we consider the problem to extend Mori contractions. Let Y be a
smooth projective variety with Picard number ρ := ρ(Y ) ≥ 2, and suppose that R ⊂
N1(Y ) is a KY -negative extremal ray of NE(Y ) such that the associated Mori contraction
π : Y → Z
is of fiber type, namely, that dimZ < dimY .
20 M.C. BELTRAMETTI, T. DE FERNEX, AND A. LANTERI
We fix a rational curve C in a general fiber of π passing through a general point
of Y , whose numerical class [C] generates R. Then we fix a normalization morphism
f : P1 → C ⊂ Y , and let
W = 〈[f ]〉 ⊂ Hombir(P
1, Y )
be the minimal family of rational curves on Y generated by [f ]. Note that, by construction,
R≥0[W ] = R≥0[C] = R in N1(Y ),
and W is a covering family on Y , since C passes through a general point of Y . Moreover,
π can be considered as the RCW -fibration of Y .
Theorem 4.1. With the above notation, assume that Y is a smooth subvariety, with
ample normal bundle, of a smooth projective variety X, such that the inclusion i : Y →֒ X
induces an isomorphism i∗ : N1(X) → N1(Y ). Let
V := 〈i∗(W )〉 ⊂ Hombir(P
be the extension of the family W to X, and assume that V is an unsplit family on X.
Then R≥0[V ] is a KX -negative extremal ray of X. Moreover, denoting by φ : X → S
the Mori contraction of this ray, there is a finite surjective morphism δ : Z → S giving a
commutative diagram
// S.
Proof. By the hypothesis and duality, the inclusion i : Y →֒ X induces a natural isomor-
phism ι : N1(Y ) ∼= N1(X). Let RX := ι(R). This is a ray in N1(X), and is contained in
NE(X), because of the inclusion ι(NE(Y )) ⊆ NE(X). Moreover, if C is a curve as above,
then adjunction formula gives
KX · C = (KY − detNY/X) · C < 0,
and thus RX is a KX -negative ray. The main point here is to prove that RX is an extremal
ray of NE(X).
We start by observing two things. First of all, we have RX = R≥0[V ] in N1(X),
and we can consider φ as the RCV -fibration of X. Moreover, if VY := 〉i
∗ (V )〈 is the
restriction of V to Y , then we have
R≥0[VY ] = ι
−1(R≥0[V ]) = R≥0[W ] in N1(Y ),
by the isomorphism ι : N1(Y ) ∼= N1(X). Therefore VY and W define the same rational
quotient, by Proposition 2.20.
Note that V satisfies the conditions (i) and (ii) of Theorem 3.1. Then, by Lemma 3.4
and the fact that V is unsplit, we have
(11) Locus(V, {0} → Y ) = X.
Indeed the lemma implies that Locus(V, {0} → Y ) is dense in X, and the unsplitness of
V implies that such locus is proper, hence closed in X.
AMPLE SUBVARIETIES AND RATIONALLY CONNECTED FIBRATIONS 21
Let D be any divisor on X whose restriction D|Y is a good supporting divisor for
R. Note that D · C = 0. Let Γ be any curve on X. Then, by Proposition 2.30, there is a
curve ΓY ⊂ Y such that [Γ] = a[ΓY ] + b[C] in N1(X), with a ≥ 0. This implies that
D · Γ = aD · ΓY + bD · C = aD|Y · ΓY ≥ 0,
since D · C = 0 and D|Y is nef. This proves that D is nef.
Since R is an extremal ray of NE(Y ), by duality it corresponds to it an extremal face
of Nef(Y ) of maximal dimension ρ − 1, and therefore we can find ρ− 1 good supporting
divisors of R whose numerical classes are linearly independent in N1(Y ). By the previous
argument, this implies that such good supporting divisors, up to numerically equivalence,
extend to divisors on X that are nef and trivial on RX , and whose numerical classes are
linearly independent in N1(X). Since there are ρ − 1 of them, and ρ = ρ(X) by the
isomorphism N1(X) ∼= N1(Y ), this implies that RX is an extremal ray of NE(X).
So, let φ : X → S be the Mori contraction of RX . By taking the Stein factorization
of the restricted morphism φ|Y : Y → S, we obtain a commutative diagram
// S,
where T is normal, ψ is surjective with connected fibers, and δ is finite. We observe that
an irreducible curve on Y is contracted by ψ if and only if it is numerically proportional
(in X, and hence in Y ) to C, and therefore if and only if it is contracted by π. This
implies that T = Z and ψ = π. Moreover, by (11), we see that Y meets every fiber of φ,
and therefore δ is surjective. This completes the proof of the theorem. �
Remark 4.2. Theorem 4.1 can be generalized, with minor changes, to the case of contrac-
tions of KY -negative extremal faces of NE(Y ).
Remark 4.3. A result analogous to Theorem 4.1 is obtained in [25, Proposition 5] in a
more restrictive context (in the setting of [25], Y is assumed to be the zero locus of a
regular section of an ample vector bundle on X, with dimX ≥ 4, and δ is proven to
be an isomorphism). Let us note that the proof of [25, Proposition 5] makes use in an
essential way of the fact that the divisor D (in our and in his notation as well) is an
adjoint divisor, i.e., a divisor of the form D = KX + A for some ample line bundle A on
X. This circumstance seems not to be true in general. The gap was pointed out by Paltin
Ionescu. Our arguments fill up that gap. A similar discussion is given in [2] in the case
Y is an ample divisor on X. In connection with Theorem 4.1 we should also mention [1,
Theorem 3.4] and [9, Lemma (1.4)].
We close this section with the following elementary property on extensions of relative
polarizations, which applies in particular to the setting of Theorem 4.1.
Proposition 4.4. Let
Z // S
22 M.C. BELTRAMETTI, T. DE FERNEX, AND A. LANTERI
be a commutative diagram of morphisms of projective varieties, and assume that i is an
embedding, that π is not a finite morphism, and that ρ(X/S) = 1. Let M be a line bundle
on X whose restriction M |Y to Y is π-ample. Then there exists an ample line bundle H
on X such that H|F ∼=M |F for every fiber F of π.
Proof. Since the numerical class of a curve in a positive dimensional fiber of π spans
N1(X/S), it follows from the relative Kleiman criterion of ampleness (see, e.g., [17, Theo-
rem 1.4]) that M is φ-ample. Then the line bundle H := M + φ∗(kA) is ample if A is an
ample line bundle on S and k ≫ 0 (see, e.g., [22, Proposition 1.7.10]), and H|F ∼= M |F
for every fiber of π. �
5. Projective bundles and quadric fibrations as ample subvarieties
The results obtained in the previous sections are here applied to classify, under
suitable conditions, projective manifolds containing “ample subvarieties” that admit a
structure of projective bundles or quadric fibrations.
Here we will adopt the following definitions of projective bundles and quadric fi-
brations. We remark that there are in fact several slightly different notions of quadric
fibrations in the literature, and the one adopted here is a little more restrictive than
others (see Remark 5.2 below).
Definition 5.1. A surjective morphism π : Y → Z between smooth projective varieties
is said to be a Pm-bundle (resp., a Qm-fibration) if π is an extremal Mori contraction and
there is a line bundle L on Y such that every fiber F of π is mapped isomorphically to
Pm (resp., is embedded as an irreducible and reduced quadric hypersurface of Pm+1) via
the complete linear system |L|F |.
Remark 5.2. With the above definition, a morphism π : Y → Z is a Pm-bundle if and
only if Y ∼= PZ(F) for some vector bundle F of rank m + 1 on Z; we can take as L the
tautological line bundle of F on Y . On the other hand, not all scrolls (resp., quadric
fibrations) in the adjunction theoretic sense (cf. [3, Sections 3.3, 14.1, 14.2]) are Pm-
bundles (resp., Qm-fibrations) in our sense. In fact, our definition of Qm-fibration does
not include conic bundles with singular fibers, since we require all fibers to be irreducible;
moreover, the assumption that π is an extremal Mori contraction implies that ρ(Y/Z) =
1, which excludes (as we will see in the proof of Lemma 5.3) those fibrations with all
fibers isomorphic to Q2 for which the fundamental group of the base Z acts with trivial
monodromy. Note that the general fiber of a Qm-fibration is isomorphic to Qm.
Lemma 5.3. Let π : Y → Z be either a Pm-bundle or a Qm-fibration, and let W be the
family of rational curves on Y generated by the lines in the fibers of π. Then W is an
irreducible family.
Proof. The assertion is clear in all cases except when π is a Q2-fibration. In this case, a
straightforward extension of the arguments in the proof of [8, Proposition (1.3.1)] shows
two things. First of all, ρ(Y/Z) 6= 1 if and only if all fibers of π are isomorphic to Q2
and the fundamental group of Z acts with trivial monodromy. Moreover, in all remaining
cases, the lines in the fibers generate an irreducible family. The fact that the base is
one-dimensional in the setting considered in [8] is not essential in the arguments. �
AMPLE SUBVARIETIES AND RATIONALLY CONNECTED FIBRATIONS 23
Definition 5.4. An embedding Pm →֒ Pn is said to be linear if the image of Pm is a
linear subspace of Pn. Similarly, an embedding of irreducible varieties F ⊂ G, with F
isomorphic to a quadric of Pm+1 and G isomorphic either to Pn or to a quadric in Pn+1,
is said to be linear if OG(1)|F ∼= OF (1).
For the reminder of this section, we consider the following setting. LetX be a smooth
projective variety, and assume that Y ⊂ X is a positive dimensional submanifold with
ample normal bundle NY/X . An easy application of a well-known theorem of Kobayashi
and Ochiai gives us the following property, in which the case when Y is a projective space
or a smooth quadric of dimension ≥ 3 is considered.
Proposition 5.5. With the above notation, assume that the inclusion i : Y →֒ X induces
an isomorphism i∗ : Pic(X) → Pic(Y ), and denote n := dimX. If Y ∼= Pm for some
m ≥ 1, then X ∼= Pn and Y is linearly embedded in X. If Y ∼= Qm for some m ≥ 3, then
either X ∼= Pn or X ∼= Qn, and Y is linearly embedded in X.
Proof. Suppose first that Y ∼= Pm. By the assumption, we have ρ(X) = 1, and the ample
generator OX(1) of Pic(X) satisfies OX(1)|Y ∼= OY (1). Observe that
d := degNY/X ≥ rkNY/X = n−m.
By adjunction, we have OX(−KX) ∼= OX(m+ 1+ d), and therefore X is a Fano manifold
of index ≥ n + 1. Then the assertion follows from Kobayashi and Ochiai’s theorem (see
e.g., [3, (3.1.6)]). The argument for the case when Y ∼= Qm with m ≥ 3 is analogous. �
Remark 5.6. The case when Y ∼= Q2 is harder to deal with under such weak hypotheses.
However, if we additionally assume the existence of an ample line bundle H on X such
that H|Y ∼= OP1×P1(1, 1), then this case cannot occur. If at the same time we relax the
hypothesis on the Picard groups, and only assume that i∗ : Pic(X) → Pic(Y ) is injective,
then it is easy to see that necessarily ρ(X) = 1, either X ∼= Pn or X ∼= Qn, and Y
is linearly embedded in X. Indeed, to exclude the case ρ(X) = 2, it suffices to apply
Theorem 4.1 with the family W being either one of the two rulings of Q2. In this way one
obtains two distinct Mori contractions of X whose relative dimensions, added together,
exceed the dimension of X, but this is impossible.
Let us now recall the following extension of the Lefschetz hyperplane theorem, es-
sentially due to Sommese [27, Proposition 1.16] (see also [19, Theorem 1.1]), which will
be used in the proof of case (b) of Theorem 5.8 below.
Proposition 5.7. Let X be a projective manifold, and suppose that Y ⊂ X is a submani-
fold defined by a regular section of an ample vector bundle on X. Suppose that dimY ≥ 3.
Then the inclusion i : Y →֒ X induces an isomorphism i∗ : Pic(X) → Pic(Y ).
We now address the case when Y is a Pm-bundle or a Qm-fibration over a smooth
projective variety. The following theorem (which includes Proposition 5.5 as a very special
case) is the main result of this section.
Theorem 5.8. Let Y be a submanifold of a projective manifold X with ample normal
bundle. Assume that Y admits either a Pm-bundle or a Qm-fibration structure π : Y → Z
over a smooth projective variety Z (see Definition 5.1), and that Y has codimension
(12) codimX Y < dimY − dimZ.
24 M.C. BELTRAMETTI, T. DE FERNEX, AND A. LANTERI
Further suppose that one of the following two situations occurs:
(a) Z is a curve and the inclusion i : Y →֒ X induces an isomorphism Pic(X) ∼=
Pic(Y ); or
(b) Y is the zero scheme of a regular section of an ample vector bundle E on X.
Then π extends to a morphism π̃ : X → Z, that is, there is a commutative diagram
eπ~~}}
Moreover, letting n := m+ codimX Y , the following holds:
• if π is a Pm-bundle, then π̃ is a Pn-bundle; and
• if π is a Qm-fibration, then π̃ is either a Pn-bundle or a Qn-fibration.
In either case, the fibers of π are linearly embedded in the fibers of π̃.
Proof. For clarity of exposition, we discuss case (a) and case (b) separately, as certain
steps require different arguments.
Case (a). Let W be the family of rational curves on Y generated by the lines in
the fibers of π. Note that W is an irreducible family by Lemma 5.3, and that π is the
RCW -fibration. By Propositions 2.6 and 2.26, Hombir(P
1,X) is smooth at the generic
point of i∗(W ). Let V = 〈i∗(W )〉 (note that V is irreducible). By Proposition 2.20, the
rational fibration defined by the restriction VY of V to Y coincides with π. Then, by
Corollary 3.13, we obtain the commutative diagram
// S,
where φ is the RCV -fibration and δ is a finite morphism of smooth projective curves.
By definition, there is a line bundle L on Y such that L|F ∼= OF (1) for every fiber
F of π. Since L extends to a line bundle on X and ρ(X/S) = 1, Proposition 4.4 implies
that there exists an ample line bundle H on X such that H|F ∼= OF (1) for every F .
If C is a line in a fiber of π and we set b = n if Y is a Pm-bundle and b = n − 1 if
Y is a Qm-fibration, then we have
(KX + bH) · C = (KY + bH|Y ) · C − detNY/X · C < 0.
It thus follows from the classification of polarized manifolds with large nef-value due to
Fujita and Ionescu (see [11, Section 1] or [14, Section 1]) that φ : X → S is either a
Pn-bundle or a fibration with fibers isomorphic to hyperquadrics in Pn+1 and relative
Picard number 1, and in fact, since dimX ≥ 3 and dimS = 1, in the second case X is a
Qn-fibration.
To conclude, observing that the general fiber of φ is a homogeneous space, we deduce
from Remark 3.9 that δ is in fact an isomorphism, and therefore we get a diagram as in
the statement by setting π̃ := δ−1 ◦ φ.
AMPLE SUBVARIETIES AND RATIONALLY CONNECTED FIBRATIONS 25
Case (b). The first step is to prove the existence of a diagram as in the statement.
We will use different arguments according to the codimension of Y . For short, let r := rkE.
Note that codimX Y = r = n−m.
If r = 1, then (12) implies that π has relative dimension m ≥ 2. Therefore in this
case we can apply a general result of Sommese [26, Proposition III], which says that π
extends to a morphism π̃ : X → Z.
We now assume that r ≥ 2. Note that in this casem ≥ 3 by (12). To extend π in this
situation, we will use the results from the previous sections. As in the proof of case (a), let
W be the irreducible family of rational curves on Y generated by the lines in the fibers of
π, and let V be the unique irreducible component of Hombir(P
1,X) containing the generic
point of i∗(W ). By Proposition 2.20, the rational fibration defined by the restriction VY
of V to Y coincides with the one defined by W , that is, with π. Then, by Theorems 3.1
and 3.6, we obtain a commutative diagram
//___ S,
where φ is the rational quotient defined by V and δ is a dominant and generically finite
map. Let G be a general fiber of φ, let F be one of the fibers of π that is contained in G,
and fix a line C ⊆ F (we can assume that G and F are smooth). Note that dimG = n.
Lemma 5.9. Assuming r ≥ 2, the family V is unsplit.
Proof of the lemma. Since the section of E defining Y in X restricts to a regular section of
E|G whose zero scheme is F , we can apply Proposition 5.7 to this setting. This implies that
the inclusion of F in G induces an isomorphism Pic(G) ∼= Pic(F ). Note that ρ(F ) = 1,
since F is either a projective space or a quadric of dimension m ≥ 3. Therefore we have
ρ(G) = 1. Let L be a line bundle on Y such that L|F ∼= OF (1), and let L̃ be the extension
of L to X. We observe that L̃|G is ample and −KG ≡ aL̃|G for some integer a, which
is easily seen to be positive. Then, since L̃|G · C = 1, we see that G is a Fano manifold
of index a. In particular, we have −KG · C = a ≤ n + 1. Note that, if X
◦ ⊆ X is as in
Theorem 2.13, then we can assume that G is a fiber of the morphism X◦ → Z, and hence
KX |G = KX◦ |G = KG. Therefore
detE · C = KY · C −KX · C = KF · C −KG · C ≤ −m+ n+ 1 = r + 1.
Since we are assuming that r ≥ 2, this implies that detE · C < 2r. We conclude that the
family V is unsplit by Remark 2.11. �
We come back to the proof of the theorem, still assuming for the moment that r ≥ 2.
Since V is unsplit by Lemma 5.9, we can apply Theorem 4.1. This implies that, in the
above diagram, both φ and δ are morphisms and δ is finite, and moreover ρ(X/S) = 1.
Note in particular that rk Im
Pic(X) → Pic(G)
= 1. We have dimG = n. We set b = n
if F ∼= Pm, and b = n − 1 if F ∼= Qm. Denoting by C a line in a general fiber of π, we
obtain
(KG + bH|G) · C = (KF + bH|F ) · C − detNF/G · C < 0,
26 M.C. BELTRAMETTI, T. DE FERNEX, AND A. LANTERI
since NF/G
∼= NY/X |F is ample of rank n−m. By applying again [11, 14] and taking into
account that ρ(X/S) = 1, we deduce that either (G,H|G) ∼= (P
n,OPn(1)) or (G,H|G) ∼=
(Qn,OQn(1)), with the second case occurring only if F ∼= Q
m. Since the general fiber G of
π̃ is a homogeneous space, we deduce from Remark 3.9 that δ is in fact an isomorphism.1
We then define π̃ := δ−1 ◦ φ.
At this point we have a diagram as in the statement for all values of r. We claim that
π̃ is equidimensional with irreducible and reduced fibers. To see this, let Gz := π̃
−1(z) for
an arbitrary z ∈ Z, and let ∆ be any component of Gz. Note that ∆∩Y is contained in the
fiber Fz := π
−1(z) of π, which is irreducible and reduced of dimension m; in particular, we
have dim(∆∩Y ) ≤ m. Since dim∆ ≥ n and ∆∩Y is defined by a section of E|∆, we have
dim(∆ ∩ Y ) ≥ dim∆− r. We conclude that dim∆ = n so that π̃ is equidimensional, and
Fz ⊂ ∆. Recalling that Fz is irreducible and reduced, and cut out on Y by Gz (scheme
theoretically), we see that Gz is also irreducible and reduced. Since the fibers of π̃ are
irreducible and reduced, we can apply the semi-continuity of the ∆-genus [10, Section 5
and (2.1)]. We conclude that π̃ is either a Pn-bundle or a Qn-fibration over Z. �
We note that all cases in Theorem 5.8 are effective.
Remark 5.10. We already listed in the introduction a series of works related to Theo-
rem 5.8. The classical setting when Y is an ample divisor is widely studied in the litera-
ture; for a survey and complete references we refer to [3, Chapter 5] and [2]. In the case
when Y has codimension r ≥ 2, we are not aware of other results of this type in which
the base of the fibration is allowed to be arbitrary.
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Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, I-16146 Gen-
ova, Italy
E-mail address: beltrame@dima.unige.it
Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City,
UT 84112, USA
E-mail address: defernex@math.utah.edu
Dipartimento di Matematica “F. Enriques”, Università di Milano, Via C. Saldini 50,
I-20133 Milano, Italy
E-mail address: lanteri@mat.unimi.it
	Introduction
	1. Conventions and basic notation
	2. Families of rational curves on varieties
	3. Extension of rationally connected fibrations
	4. Extension of Mori contractions
	5. Projective bundles and quadric fibrations as ample subvarieties
	References
