arXiv:math/0604522v1 [math.AG] 25 Apr 2006
VARIETIES WITH AT MOST QUOTIENT SINGULARITIES
JIUN-CHENG CHEN
Abstract. We generalize the well-known numerical criterion for pro-jective spaces by Cho, Miyaoka and Shepherd-Barron to varieties with at worst quotient singularities. Let X be a normal projective variety of dimension n ≥ 3 with at most quotient singularities. Our result asserts that if C · (−KX) ≥ n + 1 for every curve C ⊂ X, then X ∼= P
n
.
1. Introduction
We work over the field C. The n-dimensional projective space Pnis
prob-ably the simplest compact (projective) complex manifold. Let KPn be the
canonical bundle (the line bundle of holomorphic n-forms). It is elementary to see that the −KPn = O(n + 1). In particular, (line) · (−KX) = n + 1
and C · (−KPn) ≥ n + 1 for all curves C ⊂ Pn. This is a rather unusual
property: recall that if X is a smooth projective variety of dimension n and KX is not nef, then the Cone theorem implies that there is a rational curve
C such that 0 < C · (−KX) ≤ n + 1.
From this perspective, the anti-canonical bundle −KPn is unusually
posi-tive. It turns out that this property characterizes Pn among smooth
projec-tive varieties of dimension n.
Theorem 1.1([CMSB02] and [Ke01]). Let X be a smooth projective variety of dimension n ≥ 3. Assume that C · (−KX) ≥ n + 1 for all curves C ⊂ X.
Then X ∼= Pn.
This result was first proved by Cho, Miyaoka and Shepherd-Barron [CMSB02] and later by Kebekus [Ke01].
Since the condition
C · (−KX) ≥ n + 1 ∀ C ⊂ X
is very strong, it is natural to ask if the assumption on smoothness is nec-essary. In [CT05a], H,-H, Tseng and the author proved the characterization result assuming that X has only isolated LCIQ singularities. The main goal of this paper is to further weaken the smoothness assumption; it builds upon methods developed in [CT05a]. The precise statement of our result is the following:
Date: November 13, 2007.
Theorem 1.2. Let X be a projective variety of dimension n ≥ 3 with at most quotient singularities. Assume that
C · (−KX) ≥ n + 1 ∀ C ⊂ X.
Then X ∼= Pn.
We now explain the strategy used in [Ke01]. First consider the projective space Pn. Let p ∈ Pn be any point. Let fPn be the blowing up of Pn along
the point p. The variety fPn is a P1-bundle over Pn−1. Let E ∼= Pn−1 ⊂ fPn
be the exceptional divisor. The normal bundle of the exceptional divisor E ∼= Pn−1 is OPn−1(−1). The variety fPn is the Chow family of minimal
degree rational curves (lines) through the point p ∈ Pn.
Now consider the variety X. Take a general point x ∈ X. Denote by ˜X the blowing up of X along x. If we can prove that ˜X is the Chow family of minimal degree (with respect to an ample line bundle) rational curves through x, then we have a good chance to prove that ˜X ∼= fPn and hence
X ∼= Pn.
When X is smooth, Kebekus proved that the Chow scheme Hxof minimal
degree rational curves through a general point x ∈ X is isomorphic to Pn−1 and ˜X ∼= fPn [Ke01]. It follows easily that X ∼= Pn. One important step
in his proof is to show that X has a lot of minimal degree rational curves through x. More precisely, one needs to show that
dim Hx≥ l · (−KX) − 2 = n − 1,
where [l] ∈ Hx. Note that dim Hx ≤ n − 1 by a standard bend and break
argument. It follows that the Chow scheme Hx has the expected
dimen-sion n − 1. Kebekus then proved that the tangent map Hx → Pn−1 is an
isomorphism [Ke00] [Ke01].
When X is possibly singular, the situation is quite different: it is difficult to have the desired lower bound on the dimension of Hx. In [CT05a] (joint
work with H.-H. Tseng), we proved the characterization result when X has at worst isolated LCIQ singularities. We used the Deligne-Mumford stack X → X, twisted stable maps into X and Mori’s bend and break techniques to show the existence of a rational curve f : P1 → X such that (1) f (P1) · (−KX) = n + 1, and (2) f (P1) does not meet the singular locus of X. Using
this fact, one can show (following [Ke01]) that Hx ∼= Pn−1 and X ∼= Pn.
In [CT05a], the condition that Xsing is isolated is essential; it ensures that
we can deform a specific rational curve in X. Our methods do not apply without this assumption.
In this paper, we develop methods to study the case when X has only quotient singularities (not necessarily isolated). The main idea is quite sim-ple: instead of considering the twisted stable map directly, we consider a double cover and study the possible degeneration types. The assumption
ensures that the possible degeneration types are limited and we are able to show the existence of a minimal degree rational curve f : P1 → X which
does not meet Xsing. Once this fact is established, it is standard [Ke01]
[CT05a] to prove that the Chow family has the right dimension, i.e. n − 1, and X ∼= Pn.
A few words on using Deligne-Mumford stacks: for a singular variety X, it is difficult to prove the existence of enough rational curves via deformation theory; obtaining a lower bound of the dimension M or(P1, X) (in terms of −KX-degree) is difficult. However, if X has only local complete intersection
quotient (LCIQ) singularities, one can study representable morphisms from a twisted curve1to the stack X whose coarse moduli space is X. This provides a reasonable alternative. We can obtain a lower bound on the dimension of the space of twisted stable maps expressed in terms of the −KX-degree and
the number of twisted points, see [CT05].
The rest of this paper is organized as follows: In Section 2, we recall basic definitions on twisted curves and twisted stable maps. We also give a formula on the lower bound on the morphism space. The main proposition (Proposition 3.7) is proved in Section 3. In Section 4, we sketch the proof following [Ke01]; we do not claim any originality of these results. We also make a few remarks in that section.
Acknowledgments
Part of this research was conducted while the author was attending the JAMI conference at Johns Hopkins University. He likes to thank the con-ference organizers. He also likes to thank Dan Abramovich, Lawrence Ein and Stefan Kebekus for helpful discussions and valuable suggestions.
2. Twisted curves and twisted stable maps
Twisted curves play an important role in this paper. One of the main mo-tivations of introducing twisted curves is to compactify the space (Deligne-Mumford stack) of stable maps into a proper Deligne-(Deligne-Mumford stack [AV02]. Roughly speaking, twisted curves are nodal curves having certain stack structures ´etale locally near nodes (and, for pointed curves, marked points). For the precise definition, see [AV02], Definition 4.1.2.
Let C be a twisted curve and C its coarse moduli space.
2.0.1. Nodes. For a positive integer r, let µrdenote the cyclic group of r-th
roots of unity. ´Etale locally near a node, a twisted curve C is isomorphic to the stack quotient [U/µr] of the nodal curve U = {xy = f (t)} by the
following action of µr:
(x, y) 7→ (ζrx, ζr−1y),
1Roughly speaking, this is a one dimensional Deligne-Mumford stack with isolated
where ζr is a primitive r-th root of unity. ´Etale locally near this node, the
coarse curve C is isomorphic to the schematic quotient U/µr.
2.0.2. Markings. ´Etale locally near a marked point, C is isomorphic to the stack quotient [U/µr]. Here U is a smooth curve with local coordinate z
defining the marked point, and the µr-action is defined by
z 7→ ζrz.
Near this marked point the coarse curve is the schematic quotient U/µr.
2.1. Twisted stable maps.
Definition 2.1. A twisted n-pointed stable map of genus g and degree d over a scheme S consists of the following data (see [AV02], Definition 4.3.1):
C −−−−→ Xf πC y π y C −−−−→ Xf¯ y S.
along with n closed substacks Σi ⊂ C such that
(1) C is a twisted nodal n-pointed curve over S (see [AV02], Definition 4.1.2),
(2) f : C → X is representable,
(3) Σi is an ´etale gerbe over S, for i = 1, ..., n, and
(4) the map ¯f : (C, {pi}) → X between coarse moduli spaces induced
from f is a stable n-pointed map of degree d in the usual sense. A twisted map f : C → X is stable if and only if for every irreducible component Ci ⊂ C, one of the following cases holds:
(1) f |Ci is nonconstant,
(2) f |Ci is constant, and Ci is of genus at least 2,
(3) f |Ci is constant, Ci is of genus 1, and there is at least one special
points on Ci,
(4) f |Ci is constant, Ci is of genus 0, and there are at least three special
points on Ci.
In particular, a nonconstant representable morphism from a smooth twisted curve to X is stable.
We say a twisted stable map C → X is rational if the coarse moduli space C of C is rational.
Let Kg,n(X , d) denote the category of twisted n-pointed stable maps to
X of genus g and degree d. The main result of [AV02] is that Kg,n(X , d) is a
by Kg,n(X , d). Let β ∈ H2(X) be a homology class. The space of twisted
n-pointed stable maps f : C → X of genus g and homology class [(π ◦f )∗(C)] =
β is denoted by Kg,n(X , β). This stack is also proper [AV02].
2.2. Morphism space from a twisted curve to a Deligne-Mumfors stack. In this paper, we use both the stack of twisted stable maps and the morphism space from C to X . Roughly speaking, an element in the morphism space M or(C, X ) is a twisted stable map together with a param-eterization on the source curve C. Let Σ ⊂ C be the set of twisted points and B ⊂ C a finite set of points (twisted or untwisted). Let f : C → X be a representable morphism. When X is smooth, we have a lower bound on the dimension of M or(C, X ; f |B) near the morphism [f ].
Lemma 2.2 (= [CT05] Lemma 4.4).
dim[f ]M or(C, X ; f |B) ≥ −C·KX+n[χ(OC)− Card(B)]−
X
x∈Σ\B
age(f∗T X , x).
Remark 2.3. When X has only LCI singularities, a similar formula still holds as long as the image of C does not lie completely in Xsing (the locus
where the stack X is singular).
2.3. Lifting. Let X be a normal projective variety with quotient singular-ities and X a proper smooth Deligne-Mumford stack such that π : X → X is isomorphic over Xreg = X − Xsing and X is a coarse moduli space of X .
Let C be a smooth irreducible curve and ¯f : C → X a morphism.
We want to “lift” the map ¯f : C → X to a map C → X . In general, this is not possible unless we endow a orbicurve structure on C [AV02] Lemma 7.2.5, or pass to a finite cover C′ → C [AV02] Theorem 7.1.1.
First consider the case when the image ¯f (C) meets the smooth locus of X. Let {pi|i ∈ I} ⊂ C be the finite set of points which are mapped to the
singular locus of X, and let C0 = C \ {pi|i ∈ I}. Since X is isomorphic
to X away from the singular locus Xsing, the map ¯f |C0 : C0 → X admits
a lifting C0 → X . By [AV02], Lemma 7.2.5, there exists a twisted curve C
with coarse moduli space C, and a twisted stable map f : C → X extending C0→ X .
Now consider the case when ¯f (C) ⊂ Xsing. Let η ∈ C be the generic
point of C. Consider the morphism η ∈ C → X. After a finite extension η′ → η, there is a lifting η′ → X since X is proper. Let η′ ∈ C′ be the
smooth irreducible curve. Note that the morphism η′ → X is defined over an open set U → X . One can extend the finite morphism η′ → η to C′→ C
since C is proper. Let C′− U = {pj|j ∈ J}. Endowing stack structures on
this finite set of points, we can extend the morphism U → X to C′ → X .
3. Untwisted rational curves
In this section, we prove the existence of ”good” rational curves with −KX-degree n + 1, i.e. rational curves which do not intersect the singular
locus of X.
Consider a smooth Fano variety X. Let x ∈ X be a general point and S = {x1, x2, · · · , xk} ⊂ X be any finite set of points which does not contain
x. It is possible to find a rational curve through the point x which misses the finite set S = {x1, x2, · · · , xk} ⊂ X. This property is, however, not true
for singular varieties.
Example 3.1. Let E ⊂ P2 be a smooth elliptic curve. Let X ⊂ P3 be
the projective cone of E. The surface X has only one LCI singularity at the vertex. The surface X is Fano and all rational curves pass through the vertex.
This example shows that the existence of a rational curve which does not meet the singular locus Xsing is non-trivial.
Notation 3.2. Let X be a normal projective variety with at worst quotient singularities. Fix a proper smooth Deligne-Mumford stack π : X → X such that X is a coarse moduli space of X and π is an isomorphism over Xreg= X \ Xsing. Note that KX = π∗KX and
C · KX = C · KX
for any (twisted) curve C → X with coarse curve C [CT05]. We start with the following lemma:
Lemma 3.3. Let R ⊂ N E(X) be any KX-negative extremal ray. Then
there exists a twisted rational curve f : C → X such that (1) C has at most one twisted point, (2) the intersection number C · (−KX) ≤ n + 1, and (3)
[(π ◦ f )∗(C)] ∈ R.
Proof. This is essentially Proposition 3.1 in [CT05a]; one only needs to note that the argument goes through when the stack X has only isolated LCI
singularities.
Unless mentioned otherwise, we make further assumptions in the rest of this paper:
Assumptions 3.4. From now on we assume that X is a projective variety of dimX = n ≥ 3 with at most quotient singularities and has the property that
C · (−KX) ≥ n + 1
for every curve C ⊂ X.
Lemma 3.5. Assumptions as in Assumptions 3.4. Let R ∈ N E(X) be any KX-negative extremal ray and x ∈ X a general point. Then there exists a
twisted rational curve f : C → X such that (1) C has at most 1 twisted point, (2) x ∈ f (C), (3) [(π ◦ f )∗(C)] ∈ R ⊂ H2(X), and (4) C · f∗(−KX) = n + 1.
Proof. We only need to verify the condition (2) by Lemma 3.3. Let f : C → X be a twisted rational curve as in Lemma 3.3 and ∞ ∈ C the twisted point on C. By Lemma 2.2,
dim[f ]M or(C, X , f |∞, n + 1) ≥ (n + 1) + (1 − 1)n = n + 1.
Denote by Hf(∞) ⊂ K0,1(X , [(π ◦ f )∗C]) the stack of twisted stable maps
through the twisted point f (∞).
Abusing the notation, we also view [f ] ∈ M or(C, X , f |∞, n + 1) as an
element of Hf(∞). Note that dim[f ]Hf(∞)≥ n − 1.
Let T → Hf(∞) be any quasi-finite morphism where T is a normal irre-ducible quasi-projective variety of dimension n − 1 such that [f ] ∈ Im(T ). Consider the pull back family UT → T of twisted stable maps and the
mor-phism iT : UT → X . Forgetting the stack structures on the source curves
and on the target stack X , one obtains a family of stable maps (into X). Denote this family of stable maps by φT : UT → T and the morphism (into
X) by ¯iT : UT → X.
Note that the homology class [(π ◦ f )∗(C)] can not be written as the sum
of at least two curve classes by Assumptions 3.4. Let q = (π ◦ f )(∞). Recall that K0,1(X , [(π ◦ f )∗(C)]) → K0,1(X, [(π ◦ f )∗(C)]) is quasi-finite
[AV02]. Being the composition of two quasi-finite morphisms, the morphism T → Hf(∞)→ K0,1(X, [(π ◦ f )∗(C)]) is also quasi-finite.
Claim 3.6. The morphism ¯iT : UT → X is quasi-finite away from the
preimage of q.
Proof. This is a standard bend and break argument. Note that every fiber of the family φT : UT → T is irreducible since the homology class [(π ◦ f )∗C]
is unbreakable. It follows that no fiber of φT : UT → T is contracted by the
morphism ¯iT.
Suppose that ¯iT is not quasi-finite (away from the preimage of q). Then
there is a curve C ⊂ UT (may not be projective) such that ¯iT(C) = q16= q.
Let φT(C) ⊂ T be the image of C and C′ → φT(C) the normalization. Note
that the morphism
C′→ φT(C) ⊂ T → K0,1(X, [(π ◦ f )∗(C)])
is still quasi-finite. Pull back the family of stable maps to the curve C′.
Compactify this family of stable maps; this is a family (over a proper curve ¯
C′) of stable maps whose image contains q and q
1. This contradicts the
unbreakable assumption on the homology class [(π ◦ f )∗(C)] by Mori’s bend
and break.
By Claim 3.6, it follows that the dimension of the image ¯iT(UT) (and the
dimension of (π ◦ iT)(UT)) is n. Thus a general point x ∈ X lies on the
image of a twisted stable map which satisfies the conditions (1), (3), and
Proposition 3.7. Assumptions as in Assumptions 3.4. Let R ⊂ N E(X) be any KX-negative extremal ray. There is a rational curve f : P1 → X such
that (1) the image π ◦f (P1) does not lie in Xsing, (2) x ∈ f (P1) for a general
point x ∈ X , (3) the class [π ◦ f (P1)] ∈ R, and (4) f∗P1· (−KX) = n + 1. Proof. By Lemma 3.5, there is a twisted rational curve C → X such that (1) C has at most one twisted point, (2) x0 ∈ f (C) (x0 is a general point of X ),
(3) [(π ◦ f )∗(C)] ∈ R ⊂ H2(X), and (4) C · f∗(−KX) = n + 1.
Let β = [(π ◦ f )∗C] ∈ H2(X). We fix this homology class in the rest of
the proof.
We may assume that C does have a twisted point, denoted by ∞ ∈ C. Denote by x∞ = f (∞) the image of ∞. May assume that f (0) = x0 and
x0 ∈ π−1(Xreg).
Step 0. Double cover of C:
Recall that the coarse curve of C is P1. Choose a 2-to-1 cover h : P1 → P1
such that h(0) = 0, h(∞) = ∞ and h is ramified at {0, ∞} ⊂ P1, e. g. h(z) = z2. Choosing a suitable stack structure at ∞ ∈ P1, we can lift
h : P1 → P1 to D → C where D is a twisted rational curve. We abuse the notation and still denote the morphism on the stack level by h : D → C. Denote the stacky point on D by ∞ and the preimage of 0 ∈ C on D by 0 (an untwisted point).
The composition f ◦h : D → X is a twisted stable map into X with −KX
-degree 2n + 2. Note that the homology class [(π ◦ f ◦ h)∗(D)] = 2β ∈ H2(X).
Step 1. Bend and break:
We first show that we can deform the curve f ◦ h : D → X . Let M or(D, C, 2) be the space of of degree 2 representable morphisms from D to C. Let M or(D, C, h|{0, ∞}, 2) ⊂ M or(D, C, 2) be the subspace consisted of morphisms h1 : D → C such that h1(0) = 0 and h1(∞) = ∞. The space
M or(D, C, h|{0, ∞}, 2) has dimension 3. By Lemma 2.2,
dim[f ◦h]M or(C, X , (f ◦ h)|{0,∞}, 2n + 2) ≥ 2n + 2 + (1 − 2)n = n + 2 > 3.
Thus we can deform the morphism f ◦ h : D → X such that the image de-forms in X . This also implies that there is a morphism [f′] ∈ M or(D, X , 2β) such that f′ : D → X is birational to its image.
Choose [g] ∈ M or(D, X , 2β) such that g : D → X is birational to its image in X and g(0) = x0. Let 1 ∈ D be an untwisted point and x1 = g(1)
its image in X .
Claim 3.8. Replacing [g] ∈ M or(D, X , 2β) and the point x1 if necessary,
we may assume that x1 ∈ π−1(Xreg), x1 6= x0, and x1 does not lie on the
image of [h] where [h] ∈ K0,k(X , β), k = 1, 2 is any twisted stable map
Proof. Let K1 ⊂ K0,1(X , 2β) be the family of twisted stable maps whose
image contains x0 and x∞. View the morphism g : D → X as an element of
K1. Note that
dim[g]K1= dim M or[g](D, X , g|{0, ∞}, 2β) − dim Aut(D, {0, ∞})
≥ 2n + 2 − (1 − 2)n − 1 = n + 1.
Here we use Lemma 2.2 and the fact dim Aut(C, {0, ∞}) = 1. The image of this family is at least a surface since the image of g : D → X deforms.
Consider all possible twisted stable maps [h] ∈ K0,k(X , β), k = 1, 2 such
that the points x0 and x∞ lie on the image of h. By Lemma 3.9, there
are only finitely many such such twisted stable maps. The image of all these twisted stable maps is one dimensional (in X ). The claim follows by
choosing a suitable [g] ∈ K1.
Let K ⊂ K0,1(X , 2β) be the substack of twisted stable maps whose
im-age contains x0, x1 and x∞. Compute the dimension of the stack K ⊂
K0,1(X , 2β) at [g]:
dim[g]K = dim M or[g](D, X , g|{0, 1, ∞}, 2β) − dim Aut(D, {0, 1, ∞}) ≥ 2n + 2 − (1 − 3)n − 0 = 2.
By Mori’s bend and break, the domain curve of some twisted stable map [h] ∈ K has to degenerate to at least two irreducible components.
Step 2. Analysis on possible types of degeneration:
Since D · (−KX) = 2n + 2 and every curve has −KX-degree at least n + 1,
only two components of the domain curve are not contracted. Note that the coarse space of domain curve is a rational tree with an extra special point (coming from the original stacky point of D). It is easy to see that the domain curve can only break into two or three pieces.
Case I. The domain curve has three irreducible components.
Denote these components by D1, D2 and D3. Denote by f1 : D1 → X ,
f2 : D2 → X and f3 : D3 → X the twisted stable maps. Let D2 be the
component which intersects other two components. Note that D2 has to be
contracted.
Note that [π ◦ f1(D1)] + [π ◦ f3(D3)] = [π ◦ g(D1)] = 2β, and
(π ◦ f1)∗D1· (−KX) = n + 1 = (π ◦ f3)∗D3· (−KX).
Since β ∈ R and R is an extremal ray, the class [π ◦ f1(D1)] is a multiple of
the class [π ◦ f3(D3)]. Since they have the same −KX-degree, [π ◦ f1(D1)] =
[π ◦ f3(D3)] = β.
By symmetry, we only need to consider the following cases:
Case I-b: The stacky point ∞ ∈ D2 and {x0, x1} ⊂ f1(D1).
Case I-a. We will show that Case I-a forms a finite set. Pushing forward to X, and noting that
(1) π(x0) ∈ (π ◦ f1)(D1), π(x∞) ∈ (π ◦ f1)(D1), and
(2) π(x1) ∈ (π ◦ f3)(D3), π(x∞) ∈ π ◦ f3(D3),
it follows that there are only finitely many such stable maps by Lemma 3.9.
Case I-b. This is impossible by our choice of x0, x1, and x∞(see Claim 3.8).
Case II. The domain curve has two components, denoted by D1 and D2,
intersecting at a node, denoted by q. Consider the following two cases:
Case II-a: the node q is an untwisted point, i.e. q ∈ π−1(X reg).
Case II-b: the node q is a twisted point.
Case II-a. This is easy; one of the irreducible components has no twisted point on it.
We divide Case II-b into several subcases. It suffices to study the following subcases by symmetry:
Case II-b-1: the twisted point ∞ ∈ D1 and {x0, x1} ⊂ f1(D1).
Case II-b-2: the twisted point ∞ ∈ D2 and {x0, x1} ⊂ f1(D1).
Case II-b-3: the twisted point ∞ ∈ D1, x0∈ f1(D1) and x1 ∈ f2(D2).
Case II-b-1. This subcase is not possible by our choice of the points x1,
x0 and x∞.
Case II-b-2. In this case, D1 has one twisted point and D2 has two twisted
points. Consider f1 : D1 → X . Note that p = f1(q) ∈ X , the image of the
node q, is a stacky point on X .
Since the image of D1 contains x1 and x0 and the class β is unbreakable,
there are only finitely many such twisted stable maps thanks to Lemma 3.9. Since every such twisted stable map has only finitely many twisted points (the image only intersects the stacky locus at finitely many points), it follows that there are only finitely many such stacky points p (since p is the image of a twisted point on the source curve).
Recall that D2 has two stacky points, denoted by {q, ∞}, such that
f2 : D2 → X as an element of K0,2(X , β). Consider the set of all
possi-ble twisted stapossi-ble maps [f ] ∈ K0,2(X , β) such that π ◦ f (q) = π(p) and
π ◦ f (∞) = π(x∞). This set is a finite set by Lemma 3.10. Since (1) p (the
image of the node) is chosen from a finite set, and (2) for every p there are only finitely many such twisted stable maps [f ] ∈ K0,2(X , β), it follows that
there are only finitely many such twisted stable maps f2 : D2 → X . This
shows that Case II-b-2 also forms a finite set.
Case II-b-3.
The image of f1 : D1 → X contains x∞ and x0. There are only finitely
many such f1: D1 → X by Lemma 3.9. It follows that the set of all possible
nodes p is also finite as in Case II-b-2.
The image of f2 : D2 → X contains p and x1. By Lemma 3.9 again, the
set of all such f2 : D2 → X is also finite.
Step III. Concluding the proof:
By Step II, all bad cases, i.e. Case I-a, Case I-b, Case b-1, Case II-b-2 and Case II-b-3, form a finite set. Denote this finite set by S. Since dimK ≥ 2, we can find a proper irreducible curve T → K which is finite to its image and does not meet the finite set S. Pull back the family over K to T . It follows that Case II-a is the only possible type of degeneration in this
family. This concludes the proof.
The next two lemmas are needed in the proof of Proposition 3.7.
Lemma 3.9. Let β ∈ H2(X) be a homology class such that β is unbreakable,
i.e. it can not be written as the sum of at least two (effective) curve classes. Let x1 and x2 be two distinct points on X and x1 ∈ π−1(Xreg). Let k = 1, 2.
Then there are only finitely many stable maps [h] ∈ K0,k(X , β) such that
x1 ∈ h(C) and x2 ∈ h(C).
Proof. The proofs of k = 1 case and k = 2 case are similar; we only prove k = 1 case here. For any [f ] ∈ K0,0(X, β), the domain curve is irreducible
since β is unbreakable. By bend and break, there are only finitely many stable maps [fi] ∈ K0,0(X, β) whose image contains π(x1) and π(x2). Denote
this finite collection by I = {[fi]|i ∈ I}. Since x1 ∈ π−1(Xreg), the image
fi(P1) can only intersect Xsing at finitely many points.
Consider the composition morphism
K0,1(X , β) → K0,1(X, β) → K0,0(X, β),
where the morphism K0,1(X , β) → K0,1(X, β) is quasi-finite [AV02], and
the morphism K0,1(X, β) → K0,0(X, β) (forgetting the marked point) is
projective. Consider any twisted stable map [f ] ∈ K0,1(X , β). Let q be the
stacky point on the domain curve. Since a stacky point can only be mapped to a stacky point, the image p := f (q) lies in the stacky locus of X and π(p) ∈ Xsing. Note that one can only endow non-trivial stack structures
ways to endow stack structures on the source curve such that we can lift the stable map (into X) to a twisted stable map (into X ). This concludes the proof.
Lemma 3.10. Let β ∈ H2(X) be the homology class in Lemma 3.9. Let x1
and x2 be two points on X (not necessarily distinct). Then there are only
finitely many stable maps [h] ∈ K0,2(X , β) such that π ◦ h(0) = π(x1) and
π ◦ h(∞) = π(x2), where 0 and ∞ are the twisted points on the source curve
of [h].
Proof. Consider the forgetful map K0,2(X , β) → K0,2(X, β). Recall that this
morphism is quasi-finite and the stack K0,2(X , β) is proper [AV02]. Consider
the collection of all possible (2-pointed) stable maps [f ] ∈ K0,2(X, β) such
that f (0) = π(x1) and f (∞) = π(x2). It has only finitely many elements by
Mori’s bend and break. This concludes the proof.
The next proposition is standard:
Proposition 3.11. Assumptions as in Assumptions 3.4. There exists a twisted stable map f : P1 → X such that its −K
X-degree is n + 1 and its
image does not meet π−1(Xsing).
Proof. By Proposition 3.7, there is a twisted rational curve g : P1 → X
whose image does not lie on π−1(Xsing). Pick a point q ∈ g(P1) ⊂ X which
is not on π−1(Xsing). The dimension (at the point [g]) of the family of
twisted stable maps [f ] ∈ K0,0(X , n + 1) whose image contains q and meets
π−1(Xsing) is at most dim Xsing ≤ n − 2 (by a bend and break argument),
while the dimension (at [g]) of twisted stable maps [f ] ∈ K0,0(X , n + 1)
whose image contains q is at least n − 1 (by Lemma 2.2 and Claim 3.6). By a simple dimension count, it is easy to see that there is a rational curve
which does not meet π−1(Xsing).
Remark 3.12. From Proposition 3.11, it follows easily that for a general point x ∈ X there exists a curve [g] ∈ K0,0(X , β) whose image contains x
and does not meet π−1(X sing).
Notation 3.13. Let f : P1 → X be the rational curve in Proposition 3.7. Let β = [(π ◦ f )∗(P1)] ∈ R ⊂ H2(X). We will fix this class in the rest of this
section.
The next lemma is a simple observation:
Lemma 3.14. Assumptions and notation as in Assumptions 3.4 and No-tation 3.13. The Deligne-Mumford stack K0,0(X , β) is a projective scheme.
Proof. Let [f ] ∈ K0,0(X , β). Note that the domain curve is always
irre-ducible since there is no marked point and n + 1 is the minimal possible −KX-degree. Note that f : P1 → X is birational to its image in X (again,
is no non-trivial automorphism for any twisted stable map in K0,0(X , β).
It follows that K0,0(X , β) is a proper scheme. Since it is quasi-finite to the
projective scheme K0,0(X, β). It has to be projective.
Let [f ] ∈ K0,0(X , β) be a twisted stable map into X which does not meet
the preimage of Xsing. Let Z ⊂ K0,0(X, β) be an irreducible component
which contains [f ] and ˜Z the normalization. Consider the finite morphism K0,0(X , β) → K0,0(X, β). Let Z′ be the component of K0,0(X, β) which
contains the image of Z, i.e. stable maps of the form [π ◦ h] with [h] ∈ Z. We also take the normalization ˜Z′ of Z′. The next lemma compares these
two components. It is needed in the next section.
Lemma 3.15 ([CT05a] Lemma 3.12). The natural map ˜Z → ˜Z′ is an
isomorphism.
4. Results from [Ke00] and [Ke01] and some remarks
We return to work on the variety X, rather than the stack X . Most results in this section are taken from [Ke00] and [Ke01]. The reader can also consult [CT05a] Section 4 for more details.
Let x be a general point on X and R ∈ N E(X) a KX-negative extremal
ray. Let f : P1 → X be a twisted rational curve such that (1) P1·f (−KX) = n + 1,
(2) [π ◦ f (P1)] ∈ R, and
(3) the image f (P1) contains x and does not meet π−1(X sing).
The existence of such a curve follows from Proposition 3.11 and Remark 3.12. Fix the homology class [(π ◦ f )∗(P1)] ∈ R ⊂ H2(X)]. Let [f ] ∈ Hx ⊂
K0,0(X , [(π ◦ f )∗(P1)]) be an irreducible component of the family of stable
map through x. By choosing a suitable component (in fact, there is only one component), we may assume that
dim[f ]Hx ≥ P1· f∗(−KX) − dimAut(P1, 0) = n + 1 − dimAut(P1, 0) = n − 1.
By a standard bend and break argument (see Claim 3.6), we have dimHx=
n − 1.
Let ˜Hx→ Hx be the normalization. By Lemma 3.15, the variety ˜Hx can
be viewed as an irreducible component of the subfamily of stable maps (into X) passing through x.
Consider the diagram
Ux i x −−−−→ X πx y ˜ Hx
where πx : Ux → ˜Hx is the universal family over ˜Hx and ix : Ux → X
the universal morphism into X. Note that the preimage i−1(x) contains a
Let ˜X → X be the blowing up of X along a general point x. There is a rational map ˜ix: Ux99K ˜X lifting the morphism ix : Ux → X.
We now sketch the proof of Theorem 1.2 following [Ke01].
Step 1. The variety Ux ∼= fPn.
Let E ∼= P(TX∗|x) be the exceptional divisor of ˜X → X, the blowing up
of X along the point x. First note that ˜ix|σ∞ : σ∞ → P(T
∗
X|x) ∼= Pn−1
is a finite morphism [Ke00]. By [Ke01] Proposition 3.1, this is indeed an isomorphism. In particular, ˜Hx is smooth. Since Ux is a P1-bundle over
˜
Hx, Ux is also smooth. Note that the universal property of the blow up
implies that ˜ix : Ux → X is a morphism. Also by [Ke01] Proposition 3.1,
the normal bundle Nσ∞, Ux ∼= NE, ˜X ∼= OPn−1(−1). It is now easy to see that
Ux ∼= P(OPn−1(−1)⊕OPn−1) which is just the blowing up of Pnalong a point.
Consider the morphism ix : Ux → X and its Stein factorization:
Ux α
−−−−→ Y −−−−→ X,β
where α has connected fibers and β is a finite morphism.
Step 2. Y ∼= Pn.
It is easy to see that Y ∼= Pn since both the morphisms α : Ux → Y and
α′: Ux→ Pn contract the divisor E ⊂ Ux.
Step 3. The finite morphism β : Y → X is an isomorphism.
Write KY = β∗KX+ R where R is the (effective) ramification divisor. Note
that the β-image of lines through α(x) are curves associated with ˜Hx. For
any point [C] ∈ ˜Hx (C is a rational curve of −KX-degree n + 1), we have
C · (−KX) = n + 1 = (line) · −KPn. This implies that the divisor R is empty
(recall that any effective divisor on Pn is ample) and K
Y = β∗KX.
Suppose the degree of the morphism β is d. Take a general line l ⊂ Pn∼=
Y . Let C′ ⊂ X be the image of l (under the morphism β). The class β∗[l] = d[C′]. Since −KY = β∗(−KX), it follows that
n + 1 = l · (−KY) = l · β∗(−KX) = dC′· (−KX) ≥ d(n + 1).
This shows that d = 1 and β is an isomorphism.
Remark 4.1. There is a shorter, but less elementary, proof. It uses the main theorem in [CMSB02]. Recall the main theorem from [CMSB02]: Theorem 4.2 (=[CMSB02] Main Theorem 0.1). Let X be a normal pro-jective variety over C. If X carries a closed, maximal, unsplitting doubly dominant family of rational curves, then X is isomorphic to Pn.
Proposition 3.11 implies the existence of such a family of rational curves. Remark 4.3. If X has at worst LCIQ singularities, we can prove the same result with an extra assumption that dim Xsing < (n − 1)/2. H.-H. Tseng
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Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2370, USA
