Singular rationally connected surfaces with non-zero pluri-forms

0.45 MB·English

by  Wenhao Ou

SINGULAR RATIONALLY CONNECTED SURFACES WITH NON-ZERO
PLURI-FORMS
WENHAOOU
Abstract. ThispaperisconcernedwithprojectiverationallyconnectedsurfacesX withcanonicalsingu-
larities and having non-zero pluri-forms, i.e. H0(X,(Ω1 )[⊗m])(cid:54)={0} for some m>0, where (Ω1 )[⊗m] is
X X
thereflexivehullof(Ω1 )⊗m. Forsuchasurface,wecanfindafibrationfromX toP1. Inaddition,thereis
3 X
asurfaceY withcanonicalsingularitiesandafinitesubgroupGofAut(Y)whoseactionis´etaleincodimen-
1
sion 1 over Y such that X =Y/G. Moreover there is a G-invariant fibration from Y to a smooth curve E
0 withpositivegenussuchthatP1=E/GandH0(X,(Ω1 )[⊗m])∼=H0(Y,(Ω1)[⊗m])G∼=H0(E,(Ω1)⊗m)G.
2 X Y E

n
a
J

5 Contents

] 1. Introduction and notation 1
G
2. Vanishing theorem for Fano varieties with Picard number 1 3
A
3. Mori fiber surfaces over a curve 4
. 3.1. Some properties of fibers 4
h
t 3.2. Singularities on non-reduced fibers 5
a
4. Proof of Theorem 1.3 7
m
4.1. Source of non-zero reflexive pluri-forms 8
[
4.2. Back to the initial variety 9

1 5. Proof of Theorem 1.2 9
v 6. Proof of Theorem 1.4 11
5 References 12
1
9
0
.
1
0 1. Introduction and notation
3
1 Recall that a projective variety X is said to be rationally connected if for any two general points in X,
: there exists a rational curve passing through them, see [Kol96, Def. 3.2 and Prop. 3.6]. It is known that
v
for a smooth projective rationally connected variety X, H0(X, (Ω1 ) m)= 0 for m>0, see [Kol96, Cor.
Xi IV.3.8]. In [GKKP11, Thm. 5.1], it’s shown that if a pair (X,D) iXs k⊗lt and{X}is rationally connected, then
r H0(X,Ω[m])= 0 for m>0, where Ω[m] is the reflexive hull of Ωm. On the other hand, by [GKP12, Thm.
a X { } X X
3.3], if X is factorial, rationally connected and with canonical singularities, then H0(X,(Ω1 )[ m])= 0 for
X ⊗ { }
m>0, where (Ω1 )[ m] is the reflexive hull of (Ω1 ) m. However, this is not true without the assumption of
X ⊗ X ⊗
being factorial, see [GKP12, Example 3.7]. In this paper, our aim is to classify rationally connected surfaces
with canonical singularities which have non-zero reflexive pluri-forms.
The following example is the one given in [GKP12, Example 3.7].
Example 1.1. Let π :X P1 be any smooth ruled surface. Choose four distinct points q , q , q , q in P1.
(cid:48) (cid:48) 1 2 3 4
→
For each point q , perform the following sequence of birational transformations of the ruled surface:
i
(i) Blow up a point x in the fiber over q . Then we get two ( 1)-curves which meet transversely at x .
i i − (cid:48)i
(ii) Blow up the point x . Over q , we get two disjoint ( 2)-curves and one ( 1)-curve. The ( 1)-curve
(cid:48)i i − − −
appears in the fiber with multiplicity two.
(iii) Blowdownthetwo( 2)-curves. Wegettwosingularpointsonthefiber, eachofthemisoftypeA .
1
−
In the end, we get a rationally connected surface π : X P1 with canonical singularities such that
→
H0(X,(Ω1 )[ 2])= 0 .
X ⊗ (cid:54) { }
1
2 WENHAOOU
In fact, we will prove that every projective rationally connected surface X with canonical singularities
and having non-zero pluri-forms can be constructed by a similar method from a smooth ruled surface over
P1. We have several steps:
(i) Take a smooth ruled surface π :X P1 and choose distinct points q ,...,q in P1 with r (cid:62)4.
0 0 1 r
→
(ii) For each q , perform the same sequence of birational transformations as in Example 1.1. We get a
i
fiber surface π :X P1. The non-reduced fibers of π are π q ,...,π q .
1 1 → 1 1∗ 1 1∗ r
(iii) Performfinitelymanytimesthisbirationaltransformation: blowupasmoothpointonanon-reduced
fiber and then blow down the strict transform of the initial fiber. We obtain another fiber surface
p:X P1.
f
→
(iv) Starting from X , perform a sequence of blow-ups of smooth points, we get a surface X .
f a
(v) Blow down some chains of exceptional ( 2)-curves for X X , the end product is our surface X.
a f
− →
We will prove the following result.
Theorem 1.2. If X is a projective rationally connected surface with canonical singularities such that
H0(X,(Ω1 )[ m])= 0 for some m>0, then X can be constructed by the method described above.
X ⊗ (cid:54) { }
Note that we produce some non-reduced fibers over P1 during the process above. In fact, they are the
source of non-zero forms.
Theorem 1.3. Let X be a projective rationally connected surface with canonical singularities and having
non-zero reflexive pluri-forms. If X be the result of a MMP, then X is a Mori fiber space over P1. Let
f f
p:X P1 be the fibration. If r is the number of points over which p has non-reduced fibers, then we have
f
r (cid:62)4 a→nd
m
H0(X,(Ω1 )[ m])=H0(X ,(Ω1 )[ m])=H0(P1,O ( 2m+[ ]r))
X ⊗ ∼ f Xf ⊗ ∼ P1 − 2
for m>0.
We note that both in Theorem 1.3 and in the construction of Theorem 1.2, we meet a surface named X .
f
We will see later (in Proposition 5.3) that, by choosing a good MMP, these two surfaces are identical. On
the other hand, there is a projective surface Y and a 4 : 1 cover Γ : Y X. More precisely, we have the
→
theorem below.
Theorem 1.4. Let X be a projective rationally connected surface with canonical singularities and having
non-zero pluri-forms. Let X P1 be the fibration given by Theorem 1.3. There is a smooth curve E with
→
positive genus, a normal projective surface Y with canonical singularities such that Y is a fiber surface over
E, X = Y/G, P1 = E/G, and H0(X,(Ω1 )[ m]) is isomorphic to the G-invariant part of H0(E,(Ω1) m),
∼ ∼ X ⊗ E ⊗
where G:=Z/2Z Z/2Z whose action is ´etale in codimension 1 over Y.
×
Remark 1.5. With the notation in Theorem 1.4, Y is just the normalization of X E and we note that
×P1
Y is not rationally connected. Moreover, if rY : Y(cid:101) Y is the minimal resolution of Y, we will prove
→
that H0(Y(cid:101),(Ω1Y(cid:101))⊗m) ∼= H0(Y,(Ω1Y)[⊗m]) ∼= H0(E,(Ω1E)⊗m), and the G-invariant part is isomorphic to
H0(X,(Ω1 )[ m]).
X ⊗
Y(cid:101)
rY
(cid:15)(cid:15)
Γ (cid:47)(cid:47)
Y X
4:1cover
π(cid:48) (cid:15)(cid:15) (cid:15)(cid:15)π
E γ (cid:47)(cid:47)P1
4:1cover
Remark 1.6. Conversely,givenasurfaceY withcanonicalsingularities,afinitesubgroupGofAut(Y)whose
action is´etale in codimension 1 and a G-invariant fibration from Y to a smooth curve E with positive genus
such that E/G = P1 such that every fiber is a chain of rational curves, we will have a rationally connected
surface X =Y/G such that H0(X,(Ω1 )[ m])= 0 for some m>0.
X ⊗ (cid:54) { }
Throughout this paper, we will work over C, the field of complex numbers. Unless otherwise specified,
every variety is an integral C-scheme of finite type. A curve is a variety of dimension 1 and a surface is a
3
variety of dimension 2. For a variety X, we denote the sheaf of K¨ahler differentials by Ω1 . Denote (cid:86)pΩ1
X X
by Ωp for p N.
X ∈
For a coherent sheaf F on a variety X, we denote by F the reflexive hull of F. There is an important
∗∗
property for reflexive sheaves.
Proposition1.7([Har80,Prop. 1.6]). LetF beacoherentsheafonanormalvarietyV. ThenF isreflexive
if and only if F is torsion-free and for each open U X and each closed subset Y U of codimension at
⊆ ⊆
least 2, F(U)=j F(U Y), where j :U Y U is the inclusion map.
∼ ∗ \ \ →
If V is a normal variety, let V be its smooth locus. We denote a canonical divisor by K . Moreover, let
ns V
Ω[p] (resp. (Ω1)[ p]) be the reflexive hull of Ωp (resp. (Ω1) p). By Proposition 1.7, it’s the push-forward
V V ⊗ V V ⊗
of the locally free sheaf Ωp (resp. (Ω1 ) p) to V since V is smooth in codimension 1.
Vns Vns ⊗
Let S be a normal surface. Recall that a morphism r : S(cid:101) S is called the minimal resolution of
→
singularities (or minimal resolution for short) if S(cid:101) is smooth and K is r-nef. There is a unique minimal
S(cid:101)
resolutionofsingularitiesforanormalsurfaceandanyresolutionofsingularitiesfactorsthroughtheminimal
resolution.
Definition 1.8. Let S be a normal surface and let r : S(cid:101) S be the minimal resolution of singularities of
→
S. WesaythatS hascanonical singularities iftheintersectionnumberK C iszeroforeveryr-exceptional
S(cid:101)·
curve C.
Remark 1.9. In[KM98,Def. 4.4],Definition1.8isthedefinitionforDuValsingularities. However,by[KM98,
Prop. 4.11 and Prop. 4.20], S has only Du Val singularities if and only if it has canonical singularities and
it’s automatically Q-factorial. Thus these two definitions coincide. In this case, K is a Cartier divisor and
S
K =r K . We know all Du Val singularities, they are A , D , E where i(cid:62)1, j (cid:62)3 and k =6,7,8. For
S(cid:101) ∗ S i j k
more details on Du Val singularities, see [KM98, 4.1].
§
Definition 1.10. Letp:S B beafibrationfromanormalsurfacetoasmoothcurve. Ifthenon-reduced
→
fibersofparep z ,...,p z ,thentheramification divisor Rofpisthedivisorp z + +p z Supp(p z +
∗ 1 ∗ r ∗ 1 ∗ r ∗ 1
··· −
+p z ).
∗ r
···
LetS beaprojectiverationallyconnectedsurfacewithcanonicalsingularities,thenwecanrunaminimal
model program for S (for more details on MMP, see [KM98, 1.4 and 3.7]). We obtain a sequence of
§ §
extremal contractions
S =S S S .
0 1 n
→ →···→
Since K is not pseudo-effective, neither is K . Thus S is a Mori fiber space. we have a Mori fibration
S Sn n
p:S B. Therefore we have two possibilities: either dimB =0 or dimB =1. If dimB =0, then S is a
n n
→
Fano variety with Picard number 1. Here, a Fano variety S is a normal projective variety such that K
S
is an ample Q-Cartier divisor. In 2, we will prove that S do not have any non-zero pluri-form in this−case.
§
HencewewillconcernthecasewheredimB =1. In 3, wewillstudysomepropertiesforMorifibersurfaces
§
over a curve. In the end, we will prove Theorem 1.3, 1.2 and 1.4 successively in the last three sections.
2. Vanishing theorem for Fano varieties with Picard number 1
The aim of this section is to prove the theorem below.
Theorem 2.1. Let V be a Q-factorial klt Fano variety with Picard number 1, then H0(V,(Ω1)[ m])= 0
V ⊗ { }
for any m>0.
Before proving the theorem, we recall the notion of slopes. Let V be a normal projective Q-factorial
variety of dimension d. Let A be an ample divisor in V. Then for a coherent sheaf F, we can define µ (F)
A
the slope of F with respect to A by
det(F) Ad 1
−
µ (F):= · ,
A
rank(F)
where det(F) is the reflexive hull of (cid:86)rankF F. Moreover, let
µmax(F)=sup µ (G) G F a coherent subsheaf .
A { A | ⊆ }
4 WENHAOOU
For any coherent sheaf F, there is a saturated coherent subsheaf G F such that µmax(F) = µ (G),
⊆ A A
see [MP97, Prop. III.2.4].
Proposition 2.2. Let V be a projective normal variety which is Q-factorial, let H be an ample divisor in
V, then for any two coherent sheaves F and G on V,
µmax((F G) )=µmax(F)+µmax(G).
H ⊗ ∗∗ H H
Foraproofofthisproposition,seeforinstance[GKP12,Prop. A.16]. NowwearereadytoproveTheorem
2.1.
Proof of Theorem 2.1. We may assume that dimV >1. We will argue by contradiction. Assume that there
is a positive integer m such that H0(V,(Ω1)[ m])= 0 . Let H be an ample divisor on S.
V ⊗ (cid:54) { }
Since H0(S,(Ω1)[ m]) = 0 for some m > 0, we have an injective morphism of sheaves from O to
V ⊗ (cid:54) { } S
(Ω1)[ m]. Thus µmax((Ω1)[ m])(cid:62)µmax(O )=0. Furthermore, by Proposition 2.2, we have µmax(Ω[1])=
V ⊗ H V ⊗ H S H V
m 1µmax((Ω1)[ m])(cid:62)0.
− H V ⊗
Therefore, there is a non-zero saturated coherent sheaf F Ω[1] such that µ (F) (cid:62) 0. Observe that
⊆ V H
rank(F) < dimV, otherwise F = Ω[1] and det(F) = K . Thus µ (F) < 0 a contradiction. We have two
V ∼ V H
possibilities, either µ (F)>0 or µ (F)=0.
H H
Case 1. Assume that µ (F)>0. Since V has Picard number 1, det(F) is ample and its Kodaira-Iitaka
H
dimension is dimV. However this contradicts Bogomolov-Sommese vanishing theorem (see [GKKP11, Thm.
7.2]).
Case 2. Assume that µ (F) = 0. If G =detF, then G H(dimV 1) = 0. Since V is Q-factorial and klt,
H −
·
by [AD12, Lem. 2.6], there exists an integer l such that (G l) is isomorphic to O . Let m be the smallest
⊗ ∗∗ V
positive integer such that (G m) =O . We can construct the cyclic cover q :Z V of V corresponding
⊗ ∗∗ ∼ V →
to G, see [KM98, Def. 2.52]. Then (q G) =O . Since q is´etale in codimension 1, Z is also klt by [Kol97,
∗ ∗∗ ∼ Z
Prop. 3.16] and K = q ( K ) is ample. Thus Z is rationally connected by [HM07, Cor. 1.3 and 1.5].
Z ∗ V
And there are na−tural inject−ive morphisms (q G) (cid:44) (q Ω[rank(F)]) (cid:44) Ω[rank(F)]. Hence we have an
injection O (cid:44) Ω[rank(F)], but this contradicts∗[GK∗∗K→P11,∗ThVm. 5.1]. ∗∗ → Z (cid:3)
Z → Z
3. Mori fiber surfaces over a curve
In this section, we study Mori fiber surfaces over a curve. In the first subsection, we will give some
properties of the fibers. In the second subsection, we will classify the singularities on a non-reduced fiber.
We would like to introduce some notation for this section first. Let p:S B be a Mori fibration, where
→
B is a smooth curve and S is a normal surface with canonical singularities. Let r : S(cid:101) S be the minimal
→
resolution and p˜=p r :S(cid:101) B.
Since S is singular◦at onl→y finitely many points, p is smooth over general points of P1 and general fibers
are all isomorphic to P1. Note that a point in a smooth curve can also be regarded as a Cartier divisor and
since any two fibers of p are numerically equivalent, we have K p z = 2 and p z p z =0 for any z P1
S ∗ ∗ ∗
· − · ∈
by the adjunction formula.
3.1. Some properties of fibers.
Proposition 3.1. If we run a p˜-relative MMP, we will get a smooth ruled surface over B in the end.
Proof. Let p :Sm B be the result of a p˜-relative MMP, then Sm is a smooth surface. Since K is not
pseudo-effectSivme, neith→er is K . This implies that Sm is a ruled surface over B. S(cid:101) (cid:3)
Sm
Proposition 3.2. The support of p z is an irreducible Weil divisor for every z B.
∗
∈
Proof. Assume the opposite and let C, D be 2 distinct components in p z which meet. Then C D >0 and
∗
·
C C < 0 since p z C = 0. However, there is a positive number λ such that λC and D are numerically
∗
equ·ivalent. That is C· C >0. Contradiction. (cid:3)
·
Proposition 3.3. Let z be a point in B and let C be the support of p z. Then the coefficient of C in p z
∗ ∗
is at most equal to 2.
Proof. Let α N be the coefficient. Then 2=K p z =αK C. However, since K is a Cartier divisor,
S ∗ S S
K C Z. T∈hus 2 αZ which means α−(cid:54)2. · · (cid:3)
S
· ∈ − ∈
5
Remark 3.4. Ifp z =C ascycles,thenC issmoothandS issmoothalongC. FirstnotethatS isCM,since
∗
it is a normal surface. Then, in this case, C = p z is a reduced subscheme since it is CM and generically
∗
reduced. By the adjunction formula (see [KM98, Prop. 5.73]) and the Riemann-Roch theorem (see [Har77,
Ex. IV.1.9]), we have 2h1(C,O ) 2 = (K +p z) p z = 2. This implies that h1(C,O ) = 0 and C is
C S ∗ ∗ C
isomorphic to P1. − · −
Proposition 3.5. Let z be a point in B. If S is smooth along the support of p z, then p z is a reduced
∗ ∗
subscheme.
Proof. Let C be the support of p z. Then C is an irreducible Cartier divisor by Proposition 3.2. By the
∗
adjunction formula, we have 2h1(C,O ) 2=(K +C) C =K C <0.
C S S
− · ·
Therefore, K C = 2 = K p z, which implies that p z = C as cycles. Hence p z is a reduced
S S ∗ ∗ ∗
subscheme in S fo·r it is C−M and gen·erically reduced. (cid:3)
Proposition 3.6. There exist at most 2 singular points of S on the fiber over z B.
∈
Proof. Let t be the number of singular points over z B and assume that t>0. Then the fiber p z is non-
∗
∈
reduced. LetCbethesupportofp∗zandletC(cid:101)beitsstricttransforminS(cid:101). ThenKS(cid:101)·C(cid:101) =2−1(KS(cid:101)·p˜∗z)=−1,
for KS(cid:101) is r-nef and C(cid:101) has coefficient 2 in p˜∗z. Let E = p˜∗z−2C(cid:101). Then −1 = C(cid:101)2 = 2−1C(cid:101)·(p˜∗z−E) =
2−1C(cid:101) E. Thus t(cid:54)C(cid:101) E =2. (cid:3)
− · ·
3.2. Singularities on non-reduced fibers. In this subsection, we will give a list of non-reduced fibers of
p:S B (seeTheorem3.13). Wewillassumethatphasnon-reducedfiberover0 B,thenthereexistone
→ ∈
or two singular points in S over 0 B. We will study these two cases separately. We denote the support of
∈
p∗0 by C and its strict transform in S(cid:101) by C(cid:101). First we will treat the case of two singular points.
Proposition 3.7. Assume that there are two singular points over 0 B, then each of them is of type A .
1
∈
Proof. WiththesamenotationastheproofofProposition3.6, wehaveC(cid:101) E =2. Sincethereare2singular
·
points over 0, we can decompose E into D+D(cid:48)+R such that C(cid:101) D =C(cid:101) D(cid:48) =1, D D(cid:48) =0 and C(cid:101) R=0.
· · · ·
Then 0=p˜∗z D =2C(cid:101) D+D2+D(cid:48) D+R D =R D. By symmetry, R D(cid:48) =0. Hence R=0. (cid:3)
· · · · · ·
We will denote this type of fiber by (A +A ). We note that this type of fiber does exist by Example 1.1.
1 1
Next we will study the case of one singular point. In fact, we will prove that this isolated singularity is of
type D (i(cid:62)3 and the type D is just A ). We would like to introduce some notation first.
i 3 3
Running a MMP relative to B for S(cid:101) gives a sequence:
(cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47)
S(cid:101) Y1 Yn(cid:48)
···
p˜
qn(cid:48)
(cid:15)(cid:15)
(cid:38)(cid:38)
B
where qn(cid:48) :Yn(cid:48) B is a Mori fibration by Proposition 3.1.
→ (cid:83)
Recall the definition of dual graph. Let C = C be a collection of proper curves on a smooth surface S.
i
The dual graph Γ of C is defined as follow:
(1) The vertices of Γ are the curves C .
i
(2) Two vertices C =C are connected with C C edges.
i j i j
(cid:54) ·
Proposition 3.8. The support of p˜ 0 is a snc tree, i.e. it is a snc divisor and its dual graph is a tree.
∗
Proof. In fact, S(cid:101) can be obtained by a sequence of blow-ups from Yn(cid:48). Thus the dual graph of the support
of p˜ 0 is a snc tree. (cid:3)
∗
Proposition 3.9. The isolated singularity on the fiber over 0 B can only be of type D (i(cid:62)3).
i
∈
Proof. Let C0 =C(cid:101) and let E0 =p˜∗0−2C0. As in the proof of Proposition 3.6, we have C02 =−1, E02 =−4
and E C = 2. Since there is only one singular point over C and the support of p˜ 0 is a snc tree, we
0 0 ∗
·
can decompose E into 2C +E , where C is a reduced irreducible rational curve. Since 2C p˜ 0 = 0
0 1 1 1 1 ∗
·
6 WENHAOOU
and E2 = 4, we have C2 = 2, E2 = 4 and C E = 2. Thus the support of E intersects C at one
0 − 1 − 1 − 1· 1 1 1
or two points. If they intersect at two points, then as in Proposition 3.7, E = D+D where D, D are
1 (cid:48) (cid:48)
reduced irreducible rational curves, and we have D D =0, D C =1, D C =1. If they intersect at one
(cid:48) 1 (cid:48) 1
· · ·
points,thenwecandecomposeE into2C +E whereC isreducedandirreducible,andwehaveC2 = 2,
1 2 2 2 2 −
E2 = 4 and E C = 2. We are in the same situation as before. Hence by induction, we can decompose
2 − 2· 2
E into 2(D + +D )+D+D where D, D and all D are reduced and irreducible. Furthermore, we
0 1 i (cid:48) (cid:48) j
have D D =0,·D·· D =1, D D =1, D D =1 for 1(cid:54)j (cid:54)i 1, and D D =0 if k j >1. This
(cid:48) i i (cid:48) j j+1 j k
shows th·at the singu·lar point is·of type D ·. − · − (cid:3)
i+2
We denote these types of fibers by (D ) (i (cid:62) 3) according to their singularity. Now we will prove that
i
these kinds of fibers exist.
Proposition 3.10. Let x S be a smooth point over 0 B and let W be the blow-up of S at x with
∈ ∈
exceptional divisor E W. Let D be the strict transform of C in W. Then we can blow down D and obtain
⊆
another Mori fiber surface S B.
(cid:48)
→
Proof. We have C C = 0, K C = 1, K E = 1, E E = 1 and D E = 1. Thus K D = 0
S W W
· · − · − · − · ·
and D D = 1. Hence, by [KM98, Prop 4.10], we can blow down D and obtain another Mori fiber surface
S B·. − (cid:3)
(cid:48)
→
We can use the operation in Proposition 3.10 to construct every type of non-reduced fibers.
Proposition 3.11. If S is of type (A +A ) over 0 B then S is of type (D ) over 0 B. If S is of type
1 1 (cid:48) 3
(D ) over 0 B then S is of type (D ) over 0 B∈for i(cid:62)3. ∈
i (cid:48) i+1
∈ ∈
Proof. If the fiber is of type (A1+A1), the dual graph of the support of p˜∗0 in S(cid:101) is
1 s 2
where s represents C(cid:101).
Blow up a point only located in s, we get
1 s 2
3
Blow down successively 1, 2, s, we get S . The special fiber is of type (D ).
(cid:48) 3
On the other hand, instead of blowing down, if we continue to blow up at a point only located in 3, we
get
1 s 3 4
2
Blow down 1, 2, s, 3, we get a fiber of type (D ).
4
The same skill proves that if S has special fiber of type (D ) then S is of type (D ) over 0 B. (cid:3)
i (cid:48) i+1
∈
In fact, the fibers obtained in the proposition above are all possible non-reduced fibers with one singular
point.
Proposition 3.12. All non-reduced fibers in S can be obtained by the methods described in Example 1.1 and
Proposition 3.11.
Proof. To see this, it’s enough to describe the dual graph of the support of p˜∗0 S(cid:101). If the fiber is of type
(D ) (i(cid:62)4) then this dual graph must be ⊆
i
7
1 2 4 i
··· s
3
where s represents C(cid:101). We run a p˜-relative MMP around this fiber, then the first curve contracted must be
s since other curves are all ( 2)-curves. Now i becomes a ( 1)-curve. If we contract the curves 1,...,i 1,
− − −
we get a fiber of type (D ). Thus by induction we may now assume that the fiber is of type (D ). The
i 1 3
−
dual graph of the support of p˜∗0 S(cid:101) is as below.
⊆
1 2 s
3
In a MMP, we contract s at first, then the remaining fiber is just the same as the one in the second step of
Example 1.1. This ends the proof. (cid:3)
In the end, we obtain a table of non-reduced fibers.
Theorem 3.13. Let S be a quasi-projective surface with canonical singularities and let B be a smooth curve
such that there is a Mori fibration p:S B which has non-reduced fiber over 0 B. Let r :S(cid:101) S be the
→ ∈ →
minimal resolution and p˜be p r, then we have a table
◦
Type of fiber Dual graph
1 s 2
(A +A )
1 1
2
1 3
(D )
3
s
1 i
2 4 ···
(D )
i
s
3
where the dual graph is the one of the support of p˜∗0 S(cid:101) and s corresponds to C(cid:101).
⊆
4. Proof of Theorem 1.3
We will first prove Theorem 1.3. Let X be a rationally connected projective normal surface such that X
has canonical singularities and H0(X,Ω[⊗m]) = 0 for some m > 0. Run a MMP for X. We will get a
X (cid:54) { }
sequence of divisorial contractions
X =X X X =X .
0 1 n f
→ →···→
8 WENHAOOU
Let X be the smooth locus of X .
i,ns i
Proposition 4.1. For m N, there is an injection
∈
H0(X,(Ω1 )[ m])(cid:44) H0(X ,(Ω1 )[ m])
X ⊗ → f Xf ⊗
Proof. For any i 0,...,n , X is normal, thus by Proposition 1.7,
i
∈{ }
H0(X ,(Ω1 )[ m])=H0(X ,(Ω1 ) m).
i Xi ⊗ ∼ i,ns Xi,ns ⊗
Since X X is a divisorial contraction, X x is isomorphic to an open subset of X
i i+1 i+1,ns i+1 i,ns
→ \{ }
where x X is the image of the exceptional divisor. This gives rise to an injection
i+1 i+1
{ }⊆
H0(X ,(Ω1 ) m)(cid:44) H0(X x ,(Ω1 ) m)=H0(X ,(Ω1 ) m)
i,ns Xi,ns ⊗ → i+1,ns\{ i+1} Xi+1,ns\{xi+1} ⊗ ∼ i+1,ns Xi+1,ns ⊗
by Proposition 1.7. Then we have H0(X ,(Ω1 )[ m]) (cid:44) H0(X ,(Ω1 )[ m]). The composition of these
i Xi ⊗ → i+1 Xi+1 ⊗
injections is just H0(X,(Ω1 )[ m])(cid:44) H0(X ,(Ω1 )[ m]). (cid:3)
X ⊗ → f Xf ⊗
Let f : X X be the composition of the sequence of the MMP, then H0(X ,(Ω1 )[ m]) = 0 . Thus
→ f f Xf ⊗ (cid:54) { }
X is a Mori fiber surface over a normal rationally connected curve by Theorem 2.1. We have a fibration
f
p:X P1. Let π =p f :X P1.
f
→ ◦ →
4.1. Source of non-zero reflexive pluri-forms. Inthissubsection,wewillfindoutthesourceofnon-zero
pluri-forms on X . By Proposition 1.7, we have H0(X ,(Ω1 )[ m])=H0(U,(Ω1) m), where m N and U
f f Xf ⊗ ∼ U ⊗ ∈
is any open subset of X , the smooth locus of X , such that X U has codimension at least 2 in X .
f,ns f f f
\
Ontheotherhand,wehaveanaturalmorphismoflocallyfreesheavesonX ,(p ) Ω1 Ω1 .
f,ns |Xf,ns ∗ P1 −→ Xf,ns
Furthermore, if R is the ramification divisor of pX there exist a factorisation
f,ns
|
ρ
(p ) Ω1 ((p ) Ω1 ) O (R) Ω1
|Xf,ns ∗ P1 −→ |Xf,ns ∗ P1 ⊗ Xf,ns −→ Xf,ns.
Moreover, ρ k is injective for x in an open subset V X such that X V is a finite set of points,
x f,ns f,ns
⊗ ⊆ \
where k is the residue field of x. Thus we have an exact sequence
x
0 ((p ) Ω1 ) O (R ) Ω1 G 0,
−→ |V ∗ P1 ⊗ V |V −→ V −→ −→
where G =Ω1 /(torsion of Ω1 ) is an invertible sheaf on V, for G k is of rank 1 at every point x of
V/P1 V/P1 ⊗ x
V, where k is the residue field of x. By [Har77, Ex. III.5.16], there is a filtration over V:
x
(Ω1) m =F F F F =0,
V ⊗ 0 ⊇ 1 ⊇···⊇ m ⊇ m+1
such that F /F =G (m i) (((p ) Ω1 ) O (R )) i for every i 0,...,m .
i i+1 ∼ ⊗ − ⊗ |V ∗ P1 ⊗ V |V ⊗ ∈{ }
Proposition 4.2. With the notation above, we have a natural isomorphism
H0(X ,(Ω1 )[ m])=H0(P1,O ( 2m) (p ) O (mR)).
f Xf ⊗ ∼ P1 − ⊗ |Xf,ns ∗ Xf,ns
Proof. Forz P1 ageneralpoint,thesupportC ofthefiberp zisisomorphictoP1. WehaveG =O ( 2)
∈ ∗ |C ∼ C −
and(((p ) Ω1 ) O (R )) =O forpissmoothalongC. Thus(F /F ) =O (2(i m))fori<m.
|V ∗ P1 ⊗ V |V |C ∼ C i i+1 |C ∼ C −
Hence H0(V,F /F )=0 and H0(V,F )=H0(V,F ) for i<m.
i i+1 i ∼ i+1
On the other hand, by Proposition 1.7, we have
H0(V,((p ) Ω1 O (R )) m)=H0(X ,((p ) Ω1 O (R)) m),
|V ∗ P1 ⊗ V |V ⊗ ∼ f,ns |Xf,ns ∗ P1 ⊗ Xf,ns ⊗
and
H0(X ,((p ) Ω1 O (R)) m)=H0(P1,(p ) (((p ) Ω1 O (R)) m))
f,ns |Xf,ns ∗ P1 ⊗ Xf,ns ⊗ ∼ |Xf,ns ∗ |Xf,ns ∗ P1 ⊗ Xf,ns ⊗
which is isomorphic to H0(P1,O ( 2m) (p ) O (mR)) by the projection formula. Finally we
P1 − ⊗ |Xf,ns ∗ Xf,ns
obtain
H0(X ,(Ω1 )[ m])=H0(V,F )=H0(P1,O ( 2m) (p ) O (mR)).
f Xf ⊗ ∼ m ∼ P1 − ⊗ |Xf,ns ∗ Xf,ns
(cid:3)
We note that(p ) O (mR) is a torsion-free sheaf of rank 1 on P1, thus it’s an invertible sheaf
and there is a k |ZXfs,nusch∗thXaft,nOs (k) is isomorphic to (p ) O (mR). If this k is not less than 2m,
H0(P1,O ( 2m∈) O (k))= 0P1 and there exist non-ze|rXofr,nesfle∗xivXef,pnsluri-forms over X .
P1 − ⊗ P1 (cid:54) { } f
9
Proposition 4.3. Assume that the non-reduced fibers of p : X P1 are over z ,...,z . Then for m N,
f 1 r
m →m ∈
we have (p ) O (mR) = O ([ ](z +...+z )) = O ([ ]r), where [ ] is the integer part. In
|Xf,ns ∗ Xf,ns ∼ P1 2 1 r ∼ P1 2
m
particular, H0(X ,(Ω1 )[ m])=H0(P1,O ( 2m+[ ]r)).
f Xf ⊗ ∼ P1 − 2
Proof. SinceeverytwopointsinP1 arelinearlyequivalentandeveryirreduciblecomponentofRiscontained
in a fiber, we may assume that r =1 for simplicity. By Proposition 3.2 and Proposition 3.3, R is irreducible
m
and (p ) z =2R. Assume that (p ) O (mR)=O (k z ), we have to prove that k =[ ].
|Xf,ns ∗ 1 |Xf,ns ∗ Xf,ns ∼ P1 · 1 2
We note that γ H0(P1,O (k z )) is just a rational function on P1 which can only have pole at z with
∈ P1 · 1 1
multiplicity at most k. Its pull-back to X is a rational function which can only have pole along R with
f,ns
m
multiplicity at most 2k. Thus k is the largest integer such that 2k (cid:54)m, i.e. k =[ ] (cid:3)
2
4.2. Back to the initial variety. We have studied X and now we have to reverse the MMP and pull
f
back reflexive pluri-forms to the initial variety X. In this subsection, we will prove that H0(X,(Ω1 )[ m])=
X ⊗ ∼
H0(X ,(Ω1 )[ m]) which ends the proof of Theorem 1.3.
f Xf ⊗
Let f : X X be the composition of the sequence in the MMP and let π = p f : X P1. Assume
f
→ ◦ →
that the non-reduced fibers of p are p z ,...,p z and the ones of π are π z ,...,π z , π z ,...,π z . We
∗ 1 ∗ r ∗ 1 ∗ r ∗ 1(cid:48) ∗ t(cid:48)
note that the fibers of π : X P1 have reduced components over z ,..., z P1 since p : X P1 has
→ 1(cid:48) t(cid:48) ∈ f →
reduced fibers over these points. Thus the ramification divisor over these points will not give contribution
to non-zero reflexive pluri-forms. Our aim is now to prove that H0(X,(Ω1 )[ m])=H0(X ,(Ω1 )[ m]). To
X ⊗ ∼ f Xf ⊗
achieve this, it’s enough to prove that the fibers of π :X P1 over z ,..., z are non-reduced along each of
1 r
→
their components, i.e. the coefficient of any component in π (z + +z ) is larger than 1.
∗ 1 r
···
Proposition 4.4. Let S be a projective surface which has at most canonical singularities. Let c : S S
1
→
be a extremal contraction which contracts a divisor E to a point x. Then S is smooth at x.
1
Proof. We suppose the opposite. Let rS : S(cid:101) → S and rS1 : S(cid:102)1 → S1 be minimal resolutions, we obtain a
commutative diagram
c˜ (cid:47)(cid:47)
S(cid:101) S(cid:102)1
rS rS1
(cid:15)(cid:15) (cid:15)(cid:15)
c (cid:47)(cid:47)
S S
1
Let E(cid:101) be the strict transform of E in S(cid:101). Then KS(cid:101)·E(cid:101) = r∗KS ·E(cid:101) = KS ·rS∗E(cid:101) = KS ·E < 0. Thus E(cid:101)
must be contracted by c˜. Since E is over x, c˜(E(cid:101)) is in an exceptional divisor D of rS1 :S(cid:102)1 →S1. Let D(cid:101) be
the strict transform of D in S(cid:101). Then D(cid:101) is contracted by rS for D(cid:101) = E(cid:101). Thus D(cid:101) is a ( 2)-curve in S(cid:101). On
(cid:54) −
the other hand, since E(cid:101) is over a point of D, c˜will contract a divisor in S(cid:101)which meets D(cid:101). Hence the image
of D(cid:101) by c˜, D, has self-intersection number larger than ( 2). Contradiction. (cid:3)
−
From Proposition 4.4, every exceptional divisor of f :X X is over a smooth point of X .
f f
→
Proposition 4.5. The fibers of π : X P1 over z ,...,z P1 are non-reduced along each of their
1 r
→ ∈
components.
Proof. WiththesamenotationasProposition4.4,wedenotethesetofsingularpointsofX by x ,...,x .
f 1 u
{ }
Then X = X x ,...,x . Let X = f 1(X ) X. Since f is an isomorphism around the points
f,ns f 1 u 1 − f,ns
\{ } ⊆
f−1(x1),...,f−1(xu),theopensubsetr−1(X1) X(cid:101) canbeobtainbyasequenceofblow-upsofsmoothpoints
⊆
fromrf−1(Xf,ns)∼=Xf,ns. Thusthefibersof(π◦r)|r−1(X1) overz1,...,zr arenon-reducedalongeachoftheir
components. Hence so are the fibers of π over these points. Moreover, in each π z , there is a component
∗ i
having coefficient 2. (cid:3)
From Proposition 4.5, we obtain Theorem 1.3.
5. Proof of Theorem 1.2
WewillnowproveTheorem1.2. IfX isarationallyconnectedprojectivesurfacesuchthatX hascanonical
singularities and H0(X,(Ω )[ m]) = 0 for some m > 0 and X is the result of a MMP, then X and X
X ⊗ f f
(cid:54) { }
10 WENHAOOU
are isomorphic around the singular locus of X by Proposition 4.4. The proof of Proposition 4.5 gives us an
f
idea of how to reconstruct X from X . First we construct a surface X which can be obtained from X by
f a f
a sequence of blow-ups of smooth points. Then we blow down some chains of exceptional ( 2)-curves for
−
X X andweobtainX. Todothis, wehavetostudythestructureofexceptionaldivisorsforX X .
a f a f
→ →
Proposition 5.1. Denote a germ of smooth surface by (0 S). Let h : S S be the composition of a
(cid:48)
∈ →
sequence of blow-ups of smooth points over 0 S. Let D be the support of h 0. Then any ( 2)-curve in
∗
∈ −
D meets at most 2 other ( 2)-curves. In another word, the dual graph of D cannot contain a subgraph as
−
below such that each vertex of the subgraph corresponds a ( 2)-curve.
−
1 2 4
3
Proof. Assume the opposite. We know that we can reverse the process of blow-ups by running a MMP
relatively to S. Thus these four curves will be successively contracted during the MMP. The first one
contracted cannot be 2, since the remaining should make up a tree by an analogue result of Proposition
3.8. By symmetry, assume that 1 is first contracted. If 3 (or 4) is contracted secondly, the self-intersection
number of 2 is at least 0. If 2 is contracted secondly, a further contraction will also produce a curve with
self-intersection number at least 0. But this curve is over 0 S, it must have negative self-intersection
number by the negativity theorem (see [KM98, Lem. 3.40]). Co∈ntradiction. (cid:3)
In particular, every connected collection of ( 2)-curves in D has a dual graph as below
−
···
Moreover if we contract such a chain we will produce a singular point of type A .
i
We will now prove that it’s possible to contract such a chain.
(cid:83)
Proposition 5.2. Let C = C be a chain of ( 2)-curves as above in a smooth surface S. Then
1(cid:54)k(cid:54)i k −
there exists a morphism c:S S such that S has canonical singularities and c contracts exactly C.
(cid:48) (cid:48)
→
Proof. WenotethatK C =0foreveryk. Thusitisenoughtoprovethattheintersectionmatrix C C
S k k j
· { · }
is negative definite by [KM98, Prop 4.10]. We have C C = 2, C C =1 and C C =0 if k <j.
k k k k+1 k j+1
Hence it’s enough to prove that for any (x ,...,x ) Ri· 0 ,− · ·
1 i
∈ \{ }
2(x2+ +x2) 2(x x + +x x )>0.
1 ··· i − 1 2 ··· i−1 i
However, theleft-handsideisjustx2+x2+(x x )2+ +(x x )2, whichispositivefor(x ,...,x )
Ri 0 . 1 i 1− 2 ··· i−1− i 1 i (cid:3)∈
\{ }
Now we can conclude Theorem 1.2. For reconstructing a rationally connected surface X with canonical
singularities and having non-zero pluri-forms, we will reverse the MMP. First of all, we take a ruled surface
X over P1. By producing (at least 4) non-reduced fibers by the skill of Example 1.1 and Proposition 3.11,
0
we obtain a canonical surface X (which is isomorphic to X , the result of a MMP, as we will see below).
f(cid:48) f
Then,startingfromX ,weblowupsuccessivelysmoothpointsandwegetasurfaceX . Finally,wecontract
f(cid:48) a
chains of ( 2)-curves which are exceptional for X X . This is always possible by Proposition 5.2. We
− a → f(cid:48)
obtain X and we have a natural morphism f : X X since every curve contracted in X X is also
(cid:48) → f(cid:48) a →
contracted in X X . It remains to prove
a → f(cid:48)
Proposition 5.3. With the notation above, if we run a MMP for X, we can obtain X in the end as a
f(cid:48)
Mori firer space.
Proof. We first run a f -relative MMP for X, and we have f : Y X in the end. Let’s proof that f
(cid:48) r → f(cid:48) r
is an isomorphism. Let rY : Y(cid:101) → Y be the minimal resolution. Since KY(cid:101) is rY-nef and KY is fr-nef, KY(cid:101)
is (fr ◦rY)-nef. Thus Y(cid:101) → Xf(cid:48) is the minimal resolution of Xf(cid:48) and we have Xf(cid:48) ∼= Y since Xf(cid:48) and Y are
isomorphic around the singular locus of X . (cid:3)
f(cid:48)