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T itle T hreefolds Homeomorphic to a Hyperquadric in P4

A uthor(s ) Nakamura, Iku

C itation Hokkaido University Preprint S eries in Mathematics, 4: 0-1-B -2

Is s ue D ate 1987-06

D O I 10.14943/48864

D oc UR L http://eprints3.math.sci.hokudai.ac.jp/900/; http://hdl.handle.net/2115/45521

T ype bulletin (article)

Threefolds Homeomorphic to a Hyperquadric in p4

Iku Nakamura

#

HOKKAIDO UNIVERSITY PREPRINT SERIES IN MATHEMATICS

Author

Title

1.

Y. Okabe, On the theory of discrete KMO-Langevin equations with

reflection positivity (I)

2. Y.

Giga and

T.

Kambe, Large time behavior of the vorticity of

two-dimensional flo\..; and its 'appl ication to vortex formation

3. A. Arai. Path Integral Representation

of tile

Index

of

Kahler-Dirac

Threefolds Homeomorphic to a Hyperquadric in p4

Dedicated to Professor Masayoshi NAGATA on his 60-th birthday

By Iku NAKAMURA

Table of contents

§ 0 Introduction

§ _{1 Hyperquadrics in p4}

§ _{2 Threefolds with KX}

=

_-.3L

§ _{.3 A complete intersection ｾ＠}

=

D

n

D'

§ _{4 Proof of (.3.2)}

§ _{5 Proof of (.3 . .3)}

§ _{6 Proof of (.3.4)}

§ 7 Proof of (.3.5)

§ _{8 Proof of (.3.6)}

§ _{9 Proof of (0.1)}

-

(0 . .3)

Appendix

§ 0 Introduction. The purpose of this article is to prove

(0.1) Theorem. A compact complex threefold homeomorphic to a

nonsingular hyperquadric

0

3 in p4 is isomorphic to

0

3 if

1

H (X,OX)

=

0 and if there is a positive integer m such that dim

o

H (X,-mK

X)

>

1.

As its corollaries, we obtain

(0.2) Theorem. A Moishezon threefold homeomorphic to

0

3 is

isomorphic to

0

3 if its Kodaira dimension is less than three.

A compact complex threefold is called a Moishezon

threefold if it has three algebraically independent meromorphic

functions on it.

(0.3) Theorem. An arbitrary complex analytic (global)

deformation of

0

3 is isomorphic to

0

3.

We shall prove a stronger theorem (2.1) in arbitrary

characteristic and apply this in complex case to derive (0.1).

The above theorems in arbitrary dimension have been proved by

Brieskorn [2] under the assumption that the manifold is

kahlerian. See also [3],[9],[11] for related results. When I

completed the major parts of the present article, I received a

preprint [14] of Peternell, in which he claims that he is able

to prove the theorems (0.2) and (0.3) without assuming the

The main idea of the present article is the same as that

of our previous work [12], in which we proved the similar

theorems for complex projective space p3. However there arises

a new problem that we have never seen in [12]. See (0.4) below.

1

Let X be a complex threefold with H (X,OX)

=

0 ,K(X,-K

X)

ｾ＠ 1 (see [ 6 ) , which is homeomorphic to a nonsingular

hyperquadric

0

3. Let L be the generator of Pic X

Ｈｾ＠

Z) with L3

equal to two. Then KX

=

-3L by Brieskorn [2], Morrow [11) and

[12,(1.1»). In the same manner as in [12), we see that dim ILl

is not less than four.

Let D and D' be an arbitrary pair of distinct members of

ILl, ｾ＠ the scheme-theoretic complete intersection D

n

D' of D

and D'. Then ｾ＠ is a pure one dimensional connected closed

analytic subspace of X containing Bs ILl, the base locus of the

linear system ILl. By studying ｾ＠ and ｾ＠ d in detail, we

re

eventually prove that the base locus Bs ILl is empty. Indeed,

we are able to verify;

(0.4) Lemma. ｾｲ･､＠ is a connected (possibly reducible) curve

whose irreducible components are nonsingular rational curves

intersecting transversally and either

(0.4.1) 5t is an irreducible nonsingular rational curve, or

ｾ＠ is "a double line" with ｾ＠ _red irreducible nonsingular,

(0.4.2)

(0.4.3)

(0.4.4)

ｾ＠ is "a double line" plus a nonsingular rational curve,

rational curves ("lines") and a (possibly empty) chain of

rational curves connecting the "lines", each component of the

chain being algebraically equivalent to zero.

It turns out after completing the proof of (0.1) that the

case (0.4.3) is impossible and the chain in (0.4.4) is empty.

It follows from (0.4) that Bs ILl is empty so that the

complete intersection ｾ＠

=

D

n

D' is irreducible nonsingular for

a general pair D and D', and that dim ILl is equal to four.

Thus we have a bimeromorphic morphism f of X onto a (possibly

singular) hyperquadric in p4 associated with the linear system

ILl. It follows from Pic X ｾ＠ Z and an elementary fact about

singular hyperquadrics in p4 that the image f(X) is nonsingular

3 and that f is an isomorphism of X onto

0 .

The article is organized as follows. In section one, we

recall elementary facts about algebraic two cycles on singular

hyperquadrics in p4. In sections 2-8, we consider a threefold X

with a line bundle L such that Pic X

ｾ＠

ZL, KX

=

-3L, L3 is

positive, K(X,L) ｾ＠ 1 (see [7]). In section 2,we prove the

vanishing of certain cohomology groups. We also prove L3

ｾ＠

2

o

and h (X,L) ｾ＠ 5.

In section 3, first we state without proof five lemmas

(3.2)-(3.6) which are detailed forms of (0.4) and then by

assuming these, prove that X is isomorphic to

0

3. In sections

4-8, we study a scheme-theoretic complete intersection ｾ＠

=

D

n

In section 9, we first give a slight improvement of a

theorem in (12) and complete the proofs of (0.1) by applying the

results in sections 2-8.

Acknowledgement. We are very grateful to A. Fujiki and

H. Watanabe for their encouragement and advices.

Z

x

K(X,L)

List of notations

integers or the infinite cyclic group

complex numbers

a nonsingular threefold

L-dimension of X, L being a line bundle on X [7]

the set of base points of the linear system ILl

the q-th cohomology group of X with coefficients in

a coherent sheaf F

dimcHq(X,F)

E

(-l)qhq(X,F)

qEZ

the sheaf of germs over X of holomorphic (resp.

nonvanishing holomorphic) functions

the ideal sheaf in

Ox

defining C, resp. ｾ＠

the sheaf of germs over X of holomorphic p-forms

the canonical line bundle of X

the line bundle associated with a Cartier divisor D

the homology class of an irreducible curve C

§ 1 Hyperquadrics in p4

(1.1) We recall elementary facts about hyperquadrics in p4.

Let xi (0

ｾ＠

i

ｾ＠

4) be the homogeneous coordinate of p4,

Fv

=

v+l

\' 2 3

L xi ' Q

v

a hypersurface defined by

Fv

=

O. The hypersurface

i=O

ｑｾ＠

(v

=

1,2,3) is irreducible and Q3 (:=

0;)

only is

nonsingular.

The hypersurface

ｑｾ＠

contains a

O} and a line ｾ＠

:=

{x_O

=

xl

=

x₂

=

O}.

small open neighborhood of ｾ＠ in

(resp. U) is homotopic to q (resp. ｾＩ＠

conic q

Let U

We may

and that

.

-

Q3

n

_{x

3 = x4

.

-

₁

be a sufficiently

assume that Qi\U

au ,the boundary

of U,is an S3-bundle over the conic q. By the Thom-Gysin

sequence , we have,

0.1.1) n

=

0,2,3,5

n

=

1,4 In particular,

H

3(aU,Z) - HO(q,Z).

=

Also by the Mayer-Vietoris sequence of Qi

=

(Qi\U) U (the

closure of U), we have,

0.1.2) n

=

0,2,4,6

n

=

1,3,5 By (1.1.1) and (1.1.2), we have,

(1.1.3)

ｈＴＨｑｾＬｚＩ＠

ｾ＠

H

3(aU,Z)

ｾ＠

HO(q,Z)

ｾ＠

Z.

(1.2) Lemma. There is a Weil divisor on

ｏｾ＠

which is not an

3 3

integral multiple of a hyperplane section H of Q

1 in H4(Q1'Z),

Proof. Let a

=

[a

O,a1,a2] be a point of the conic q, Da

=

the closure of {[aO,a1,a2,x3,x4] E p4 ; x

3,x4 E C}. Then by 3

(1.1.3), H

=

2Da in H

0.3) Lemma. Let Q b,e a quadric surface

ｑｾ＠

n

{x

4

=

O}

contained in

ｑｾＮ＠

Then

ｈＴＨｑｾＬｚＩ＠

ｾ＠

H

2(Q,Z)

Ｈｾ＠

Z ffi Z) and H2(Q,Z)

is generated by fibers of two rulings via the isomorphism of Q

wi th pl x pl.

Proof. Similar to the above. q.e.d.

(1.4) Remark. In arbitrary characteristic, any singular

hyperquadric in p4 is a cone over a hyperplane section of it,

whence i t has a Weil divisor which is not (algebraically

§ 2 Lemmas

Our first aim is to prove the following

(2.1) Theorem. Let X be a compact complex threefold or a

complete irreducible nonsingular algebraic threefold defined

over an algebraically closed field of arbitrary characteristic

1

L ｾ＠ line bundle on X. Assume that H (X,OX)

=

0, Pic X

=

ZL,

L3

>

0, KX

=

-3L, K(X,L)

ｾ＠

1. Then L3

=

2 and X is isomorphic

to a nonsingular hyperquadric in p4.

Compare [2 J, (8).

Sections 2-8 are devoted to proving (2.1). Throughout

sections 2-8, we always assume that X is a compact ｣ｯｾｰｬ･ｸ＠

threefold satisfying the conditions in (2.1). Our proof of

(2.1) is completed in (3.8) by assuming (0.4), or more

precisely, (3.2)-(3.6).

(2.2) Lemma.

(m+1)(m+2)(2m+3)/6.

3 2

Proof. We see h (X,OX)

=

0, X(X,OX)

=

1 + h (X,OX) ｾ＠ 1 and c1c 2

=

24X(X,OX)

ｾ＠

24, X(X,mL)

=

X(X,Ox) +

ｭＨ｣ｾ＠

+ c₂)L/12 + m2c₁L2/4

+ m3L3/6. Assume L3

=

1 to derive a contradiction. Let c₁c₂

=

24a, a ｾ＠ 1. Hence c

2L

=

8a by L3

=

1. We also see that X(X,L)

=

(5+5a)/3, whence 1 + a

₌

o

mod 3 and a ｾ＠ 2. Let a

=

3b + 2, b

ｾ＠

o.

Then X(X,2L)

₌

7b + (21/2), which is absurd. Consequently

L3 _{ｾ＠ 2 and X(X,mL) ｾ＠ (m+1)(m+2)(2m+3)/6 by c}

(2.3) Lemma. hO(X,L)

ｾ＠

5.

Proof. The same proof as in [11,(1.5)] works by taking d = 3,

X(X,L) ｾ＠ 5 instead of d ｾ＠ 4 and X(X,L) ｾ＠ 4. q.e.d.

(2.4) Lemma. Let D and D' be distinct members of ILl, ｾ＠ = D

n

D' the scheme-theoretic intersection of D and D'. Then we have,

(2.4.1)

(2.4.2)

(2.4.3)

(2.4.4)

( 2 . 4 . 5 )

Proof.

Hq(X,-mL)=O for ｱ］ＰＬＱＬｭ＾Ｐ［ｱ］ＲＬＰｾｭｾＳ［ｱ］ＳＬＰｾｭｾＲＬ＠

Hq(D,-mLD)=O for ｱ］ＰＬｭ＾Ｐ［ｱ］ＱＬＰｾｭｾＲ［ｱ］ＲＬｭ］ＰＬＱＬ＠

°

1

H ＨｾＬＭｌｾＩ＠

=

0, H ＨｾＬｯｾＩ＠

=

0,

HO(X,OX)

ｾ＠

HOCD,OD)

ｾ＠ ｈｏｃｾＬｏｾＩ＠

ｾ＠

C,

H3 (X,-3L)

ｾ＠

H2 (D,-2L

D)

ｾ＠ ｈＱＨｾＬＭｌｾＩ＠

ｾ＠

C.

The same as in [11,(1.7)J by using an exact sequence

ｏｾ＠

_°

[11,(1.5.1) and (1.6)].

q.e.d.

(2.5) Corollary.

§ 3 A complete intersection ｾ＠ = D

n

D'

Let X, L be the same as in section 2.

(3.1) Lemma. Let D and D' be distinct members of the linear

system ILl, ｾ＠ := D

n

D' the comnlete intersection of D and D'.

Let ｾｲ･､＠ = Al + ... + As be the decomposition of ｾｲ･､＠ into

irreducible comnonents. Then

(3.1.1) each A. is a nonsingular rational curve with LA. ｾ＠ 2,

- - - - J J

(3.1.2) if there is an irreducible component A. with LA. = 2,

1 1

then LA. ｾ＠ 1 for j ｾ＠ i.

- - - - J

1

Proof. By (2.4.3), H ＨｾＬｏｾＩ＠ = 0. Hence H (A.,OA ) = 1

°

for any

J j

j , whence A. is a nonsingular rational curve. In view of (2.4.5)

J

1 1

, h ＨｾＬＭｌｾＩ＠ = 1, whence h Ｈｾ､ＬＭｌｾ＠ ) ｾ＠ 1.

re red Therefore

s 1

L

h (A.,-L

A ) =

i=1 1 i

s

L

hO(A.,OA (-2+LA.»

ｾ＠

1.

i=1 1 i 1

The assertions

are therefore clear. See [11,(2.3)]. q.e.d.

In the subsequent sections 4-8, we shall prove the

following five lemmas;

(3.2) Lemma. Let ｾ＠ = D

n

D' be the complete intersection in

(3.1). Assume that there is an irreducible component C of ｾｲ･､＠

with LC ｾ＠ 2. Then

(3.2.1) LC = 2 and ｾ＠ is an irreducible nonsingular rational

(3.3) Lemma. Let ｾ＠

=

D

n

D' be the complete intersection in

(3.1) . Assume that there is an irreducible component C of ｾ＠ red with LC

=

1 such that ｾ＠ is nonreduced anywhere along C.

(resp. IC) be the ideal sheaf of Ox defining ｾ＠ ( resJ2.. C). Then

2 2 ...

°C(-l). I +12/12 ...

ｉｾＫｉｃＯＱ｣＠ _°c or _{l i}

ｾ＠ C C °c , then

C3.3.1) ｾ＠ _red is an irreducible nonsingular rational curve,

isomorphic to C,

(3.3.2) ｾ＠ is "a double line", to be precise, at any point P of

C, the ideal sheaf _{Ｑｾ＠ (resp. Ic)} is aiven by;

Ｑｾ＠

=

Ox px _{, -} + _°x,PY 2 ,

IC

=

°x,px + °x,pY

for suitable local parameters x and y at p,

(3.3.3) _IC :J _ｉｾ＠ :J _I 2

C'

2

Ic/IC

-

_°c EB 0cC-1), _{ｉ｣Ｏｉｾ＠}

-

°c(-1), ＱｾＯｉ｣＠ 2 ...

°C·

(3.4) Lemma. Let ｾ＠

=

D

n

D' be the complete intersection in

(3.1). Assume that there is an irreducible component C of ｾｲ･､＠

with LC

=

1 such that ｾ＠ is nonreduced anywhere along C. Assume

and that if ｾ＠ _re d is reducible, then

----meets an irreducible component C' of ｾ＠ d not contained in Bs re

C

ILl. Then ｾ＠ is a double line plus a nonsingular rational curve

C'. To be more precise,

(3.4.1) ｾ＠ d is the union of C and C' with LC

=

1, LC'

=

0 re

, the curve C intersecting C' transversally at a ｵｮｩｱｵｾ＠ pOint PO'

(3.4.2)

I _C

=

°

_X,p x +

°

Y

o

X,p

o

'

I C'

=

Ox x + Ox z

,PO ,PO

for a local Darameter system x,y and z at Po and except at PO' ｾ＠

is a double line ｡ｬｯｮｾ＠ C in the sense of (3.3.2), and reduced

along C',

(3.4.3)

(3.5) Lemma. Let ｾ＠

=

D

n

D' be the complete intersection in

(3.1). Assume that ｾ＠ is reduced at a point of an irreducible

component Co of ｾｲ･､＠ with LC

O

=

1 and that Co intersects an

irreducible comDonent C' of _- ｾ＠ _re d not contained in Bs ILl. Then,

(3.5.1)

(3.5.2)

ｾ＠ is reduced everywhere,

there exist another irreducible component C of ｾ＠ with

m

LC _m

=

1 and a chain of irreducible components C. of _- ｾ＠ with LC.

=

J - J

o (1 ｾ＠ j ｾ＠ m-l) such that ｾ＠ is the union of C. (0 ｾ＠ j ｾ＠ m), the

J

Dair C

j and Ck (j

<

k) intersect iff j

=

k-l. ｾ＠ j

=

k-l, then

C. 1 and C. intersect at a unique point p. (1 ｾ＠ j ｾ＠ m)

J- - - J J

transversally, to be precise,

(:= the completion of 0₀ )

=

C[[x,y,z))/(x,yz),

x.,p.

J

for suitable local Darameters x,y,z at P_j ,

2 0c EB 0C(-I) (C = CO'C_m)

(3.5.3) Ic/Ic

= {

0c (1) EBOC ( 1) or _{0c ( 2HBO C}

(C = C1' . . . 'C_{m-1 )}

(3.6) Lemma Let ｾ＠

=

D

n

D' be the complete intersection in

ｾｲ･､＠ is reducible, then C intersects an irreducible component C'

of ｾｲ･､＠ not contained in Bs ILl.

From (3.2)-(3.6), we infer the following

(3.7) Lemma. The linear system ILl is base point free and

dim ILl

=

4, L3

=

2.

Proof by assumina (3.2)-(3.6). In view of (2.3), we are able to

choose distinct members D and D' from ILl. Let ｾ＠

=

D

n

D' be the

complete intersection. _{Let ｾｲ･､＠}

=

_{Al + •••} + A be the s

decomposition into irreducible components. Then ｣ｬＨｾＩ＠

=

nlcl(A l )

+ ••• + nscl(A

s) E H2(X,Z) for some ni

>

°

(see [11,(2.1)]). Since L3

=

ｌｾ＠

=

n_lLA

1 + + nsLAs' there is at least a component A. with LA.

>

0.

1 1 We see that there are only three

cases;

Case 1. ｾ＠ contains an irreducible component C with LC ｾ＠ 2, red

Case 2. ｾｲ･､＠ contains no irreducible components C' with

LC' ｾ＠ 2, but contains an irreducible component C with LC

=

1

along which ｾ＠ is nonreduced anywhere,

Case 3. ｾｲ･､＠ contains no irreducible components C' with

LC' ｾ＠ 2, but contains an irreducible component Co with LC

O

=

1

such that ｾ＠ is reduced at a point of CO'

By (3.2), ｾ＠ is isomorphic to C. By (2.6), Bs ILl =

Bs

ｉｌｾｉＮ＠

Since L3

=

ｌｾ＠

=

LC

=

2, we have

ｌｾ＠

=

ｏｾＨＲＩＬ＠

so that

ｉｌｾｉ＠

is base point free. Consequently ILl is base point free and

Case 2. First we assume that ｾ＠ d is irreducible. By (3.3) re

and (3.4), ｾｲ･､＠ is isomorphic to C and ｉ｣Ｏｉｾ＠ ｾ＠ 0C(-l). Hence we

have an exact sequence,

°

ｾ＠

whence

°

follows that

°

is exact. Hence ILl is base point free.

0,

°

is exact.

HOCC,OCCl»

H1(C,OC(1»)

ｾ＠

°

Moreover hOCX,L)

=

The intersection number L3

=

ｌｾ＠

=

2 because

=

2s + 1. In this case, the proof of C3.7) is

complete.

It

2 +

Next we consider the case where ｾｲ･､＠ is reducible. Then

by C3.4) and C3.6), ｾ＠ is a double line plus a nonsingular

rational curve C' ,whence ｣ｉｃｾＩ＠

=

2cICC) + cICC') and ｌｾ＠

=

2. We 2

12 = _{0CC-l)+I C via the} define a subsheaf 12 of IC by

isomophism

ｉ｣Ｏｉｾ＠

ｾ＠

0CffiOcC-1). Let p

=

C

n

c' . We note that with

the notations in (3.4), 1₂,p (:= the stalk of 12 at p) = 0x,px +

2 Ox _,p y .

°

ｾ＠

Then we have exact sequences;

°

ｾ＠

ｾ＠ _{°c,ffiCO X/I 2 )CL)}

0C(l) ｾ＠ (OX /I 2)(L)

0,

because IC/I2

=

0C. We see that a subspace H COC,) ffi

°

°

0

H (CI

C/I2)CL» of H COC,) ffi HO(COx/I2)CL» is mapped onto

0X/IC,+I2 by the natural homomorphism.

°

h ＨｾＬｌｾＩ＠ + 2

=

5, Bs ILl

=

Bs ｉｌｾｉ＠

=

ｾＮ＠

proof of C3.7) in Case 2.

Therefore hOCX,L)

=

Case 3. By (3.5) and (3.6), ｾ＠ is reduced everywhere and ｾ＠

=

Co

+ ••• + C

m with LCO

=

LCm

=

1, LCj

=

°

(1

ｾ＠

j

ｾ＠

m-1). Then L3

=

ｌｾ＠

=

L(C

O +

...

+ C ) m

=

2. Consider an exact sequence,

°

-+ ｏｾＨｌＩ＠ -+

_0c

_{(l)®OC ffi . . . ffiO}

c

ffiO

c

(1) -+ em -+ 0.

°

1 m-1 m

It follows from this that hO(X,L)

=

2 + h

°

ＨｾＬｌｾＩ＠ = 5, and that

ILl is base point free.

Thus we complete the proof of (3.7). q.e.d.

(3.8) Completion of the proof of (2.1) by assuming (3.2)-(3.6).

Let X be a compact complex threefold with a line bundle L

satisfying the conditions in (2.1). By (3.7), we have a

bimeromorphic morphism of X onto a hyperquadric in p4. The image

f(X) endowed with reduced structure is one of

ｏｾ＠

(v

=

1,2,3).

)I(

We note Pic X

=

ZL

=

Z[f H), where H is a hyperplane section of

)I( )I(

f (X) and [ f H] is the line bundle associated with f H. If we are

given a Weil divisor (an analytic two cycle) E of f (X), then f )I( E

)I(

is a Cartier divisor of X and E = f)l( (f E) because f is

)I(

bimeromorphic. Since [f E] is an integral multiple of L, any

Weil divisor of f(X) is homologically (algebraically) equivalent

to an integral multiple of H [3, Theorem 1.4). Hence f(X) #

0

3

1,

ｯｾ＠

in view of (1.2) and (1.3). We note that over an

algebraically closed field of arbitrary characteristic, any

singular hyperquadric in p4 has a Weil divisor which is not an

integral multiple of a hyperplane section. Consequently f(X)

=

0

3.

ｓｩｮ｣･ｾＩｉＨ＠

is an isomorphism of Pic X onto Pic

0

3 (= Z[H),

Before closing this section, we prepare three lemmas for

sections 4-8.

(3.9) Lemma. J,..et.Q.

=

D

n

D' be the complete intersection in

(3.1), C an irreducible component of .Q.red' IC the ideal sheaf of

2 2 n

Ox defining C, clcrc/Ie) = s E H (C,Z) (= Z). Then X(X,ox/1c)

=

n(n+1)(sn-s + 3)/6, s

=

-3LC+2.

Proof. The first assrtion is clear from Riemann-Roch for C

=

pl. Next consider an exact sequence,

o

Then we have s

=

Q1

C

=

-3LC + 2

o.

q.e.d.

(3.10) Lemma. Let.Q. and C be the same as in (3.9). Let ｾ＠

be the natural homomorphism induced from

the inclusion of I.Q. into IC. Then ｾ＠ is injective everywhere on C

iff .Q. is reduced at a point of C.

Proof. We note that

ＨｉＮｑＮＯｉｾＩＰＰ｣＠

- 0c(-L)ffiOc(-L) is locally free,

hence torsion free. Therefore the following conditions are

equivalent to each other;

(3.10.1) _ｾ＠ is injective everywhere,

(3.10.2) _ｾ＠ is injective at a pOint q of C,

(3.10.3) ｃｯｫ･ｲＨｾＩ＠

₌

0 at a point p of C,

(3.10.4) _{I.Q. +} 12

₌

_IC

C at a point p of C,

(3.10.5) _1.Q.

₌

_{IC at a point p of C.}

(3.11) Lemma. Let I and I' Ｈｾ＠ Ox) be ideal sheaves of OX'

SUPpose that I C It and h1(OX/I)

=

0, dim supp(Ox/I)

ｾ＠

1. Then

h1(OX/I')

=

0 and X(Ox/I')

ｾ＠

1.

§ 4 Proof of (3.2)

We apply a method of Mori [9,pp. 167-170].

Assume that C is an irreducible component of ｾ＠ with LC red

ｾ＠

2. Then by (3.1.1), we have LC

=

2. Then

ＨｉｾＯｉｾＩＰＰ｣＠

-°C(-2)ffiO

C(:-2).

1

Since C

=

P , by a theorem of Grothendieck, we

2 express Ic/IC =

(4.1) Lemma. _ｉｾ＠

rt

12 _C

Proof. Suppose _ｲｾ＠ C _Ic· 2 Hence h 1 (O 112) = 0 by (2.4.3).

X C

Hence X(OX/IC) 2 ｾ＠ 1. However by (3.9), X(OX/IC) 2 = s+3 = -1

because s

=

- 4 . This is a contradiction. q.e.d.

In view of (4.1), we have a nontrivial natural

homomorphism We shall prove

C4.2) Lemma. ｾ＠ is injective.

Proof. Suppose not. Then both Ker ｾ＠ and 1m ｾ＠ are torsion free

sheaves of rank one, hence locally 0C-free. By a theorem of

Grothendieck, we express Ker ｾ＠

=

0CCc), 1m ｾ＠

=

0CCd) for some

c, d E Z. Then we have an exact sequence,

o.

Hence c + d

=

-4, b ｾ＠ c ｾ＠ -2 ｾ＠ d ｾ＠ a. Now we shall prove b

=

d

Chence a

=

b

=

c

=

d

=

-2). Assume b

<

d to derive a

contradiction. Then since HomO (OcCd),OCCb»

=

0, the sheaf C

2

2 2

splitting of IC/IC is uniquely determined in Ic/IC Define a

2

subsheaf I of IC by I

=

0c(a) + IC' Then we see readily that IC

::J I :J 15l.'

ｉＯｉｾ＠

:: 0C(a), Ic/I :: 0C(b). By H1(05l.) = H1(OX/I 5l.) = 0, we have,

1 ｾ＠ X(Ox /I )

=

X(OX/lc) + XOc/I)

=

2 + b,

whence b ｾ＠ -l. This contradicts b ｾ＠ c ｾ＠ -2. Hence a

₌

b

=

c

=

d

=

-2. Next we let J

=

1m ct> + 12 _C = _15l. + I_C2 ' Then J :J

15l.' Ic/J :: 0c(-2). Therefore

1 ｾ＠ X(OX/J)

=

X(Ox/Ic) + X(OC(-2»)

=

0,

which is a contradiction. q.e.d.

(4.3) Completion of the proof of (3.2). By (4.2), we have

the exact sequence,

o

Therefore -2 ｾ＠ a, -2 ｾ＠ b, whence a

=

b

=

-2. Hence

:: IC/Ic2. Let p be a point of X, In _.>l.,p (resp. IC _,p ) be the stalk

2

of In (resp. Ic) at p. Then In + Ic

.>l. .>l.,p ,p

=

I for any point p

C,p

of C, whence In

=

IC . This shows that 5l. is isomorphic to C

.>l., P , P

anywhere on C. Since 5l.

red is connected by (2.4.4), ｾ＠ is

isomorphic to C.

§ 5 Proof of C3.3)

C5.1) Lemma. Assume that C is an irreducible component of

ｾｲ･､＠

with LC

=

1. Then

ｉ｣Ｏｉｾ＠

- 0c ffi 0cC-1).

1 2

Proof. Since C

=

P , we express Ic/Ic

=

0cCa)ffiOcCb), a ｾ＠ b.

Then by C3.9), a + b

We shall show a

=

0, b

=

-1. We assume b ｾ＠ -2 to derive a

contradiction. Consider the natural homomorphism cI>

.

.

2

( I ｾＯｉ＠ ｾｈＹＰｃ＠ ｾ＠ IC/lc' 2 When b ｾ＠ -2, 1m cI> is contained in °c Ca )

= °cCa)ffi{O} . Let I =

ｏ｣ｃ｡ＩＫｉｾＮ＠

_{Then IC} :::J I :::J _{ｉｾ＠ and IC /I}

-0CCb). Hence by C3.11),

1 ｾ＠ XCOX/I)

=

XCOX/IC) + XCIC/I)

=

2 + b.

This is a contradiction. Hence a

=

0, b

=

-1. q.e.d.

In what follows, we assume that C is an irreducible

component of ｾ＠ d with LC

=

1 along which ｾ＠ is nonreduced

re

anywhere.

2

C5.2) Lemma. ｉｾ＠

rz.

IC'

2

Proof. Assume ｉｾ＠ C I_C'

I and

ｉｾＯｉ＠

=

ｃｉｾＯｉｾＩｾｏｃ＠

hOCOx/I)

=

1, h1CO x/I)

I + 14/14 ｾ＠

C C

Let I

=

ｉｃｉｾＮ＠ Then ｉｾ＠ :::J I :::J ｉｾ＠ 2 , IC 3 :::J

=

°c C- 1 )ffiO_c C-1). Therefore by C2.4),

=

0. Consider the natural inclusion

Then 1m l is contained in 0CffiOc(-1)ffiOCC-2) because the following

natural homomorphism is surjective,

4 4 4 4 3 4

ｾ＠ ｉ｣ｉｾＫｉｃＯｉｃ＠

=

I+IC/IC (C IC/IC)

Let I'

=

ＰｃｦｦｩｏｃＨＭＱＩｦｦｩｏｃＨＭＲＩＫｉｾＮ＠

Th n 1_{e C} 3

ｾ＠

I'

ｾ＠

I, 1_C 311'

ｾ＠

1

0C(-3). By h (OX/I)

=

0 and (3.11), we have,

1

which is a contradiction.

=

ｘＨｏｸＯｉｾＩ＠

+ X(OC(-3»

=

0

2

Hence I.Q.

rz.

IC' q.e.d.

(5.3) Completion of the proof of (3.3). We consider the

2 2

natural homomorphism ¢ (I.Q./I.Q.)00

C ｾ＠ IC/lc' By (5.2),

1m ¢ is not zero. Since 1m ¢

Ｈｾ＠

I +12/12) is a subsheaf of a _{.Q. C C}

torsion free sheaf _IC/IC2, l·t l'S 1 _oca 1 1 0 f _{y C- ree.} Since .Q. is

nonreduced along C, Im ¢ is of rank one by (3.10). Here we may

set 1m ｾ＠

=

0c(c) for some c E

Z.

Then c

=

0 or -1 because

2

( I .Q. I I .Q. ) 00 C , . "

0c(-1)ffiO

c(-1). In view of our assumption in

(3.3),

= 1m ¢

= 0, EV 1m ¢ ｾ＠

ｾ＠

°C·

being

0C' Ker ¢ ｾ＠ 0C(-2). Let E

=

Coker ¢ ｾ＠ 0C(-1), F

Then we may view

ｉ｣Ｏｉｾ＠

=

E ffi F because H1(C,Ev0F)

the dual of E. So we consider again the

homomorphism ｾ＠ as,

F

Let p be an arbitrary point of C. Then there are two

generators

x

,y _{of IC ' and two generators f, g of I .Q.,p such}

,p

that cj>(f) =

x,

¢(g) = 0,

x

mod 12 (resp. y mod 12 )

C,p C,p

generates F (resp. E) . Since f =

x

mod _IC 2 _{,p '} i t is easy to see

that f and y generates I _{C,p over Ox ,p ' so that we may take f}

instead of

x.

Then by deleting an Ox -multiple of

x

from g, we

,p

may assume g

₌

Bym for some B E

°

X,p and m

>

0, the restriction

of B to C being not identically zero. Thus we obtain local

parameters x and y E IC and B E O , m

>

0 such that

m ｉｾＬｰ＠

=

0X,px + 0x,pBy

where the restriction Bc of B to C is not identically zero. The

integer m is uniquely determined by the point p, but it is

independent of the choice of p E C. We note that m ｾ＠ 2 because

2 g E IC _,p .

Let ｾ＠

=

{U.} be a sufficiently fine covering of an open

J

neighborhood of C by Stein (or affine) open sets U .. Then by

J

(5.3.1), we have x. E r(U.,IA ) ,y. E r(U.,I_c )' B. E r(u.,Ox)

J J x J J J J

such that

(5.3.2) r(Uj,I_c ) = rcu. ,Ox)x. + r(u. ,OX)Y.

J J J J

ｲＨｕｪＬｉｾＩ＠ = r(u., 0X)x. + r (U . , Ox ) B . y .. m

J J J J J

Since 2 F

EB E

"-IC/IC = _°c EfJ 0CC-l), we may assume that

(5.3.3) x. = x mod 12 _{ｾｪｫｙｫ＠} mod 2

k , y.

=

IC'

J C J

1

*-where ｾｪｫ＠ stands for the one cocycle LC

=

0C(l) E H (C,OC)·

Note that the second equation in (5.3.3) does make sense.

m m

Hence DjBjyj and DkBkYk E Ker

¢

Ｈｾ＠ 0C(-2» are

identified iff (we may assume that)

(5.3.4)

This shows that

( 5 . 3 . 5 )

In particular, Bc := {B

j1c; Uj E ｾｽ＠ is a nontrivial

o

element of H (C,Oc(2-m». This is possible only when m

=

2 and

BC is a nonzero constant. Consequently ｾ＠ d is nonsingular re

anywhere on C, and it is isomorphic to C because it is connected

by C2.4.4). Moreover ｾ＠ is "a double line" in the sense that at

I

C,p

I

.Q.,p

=

Ox _{, p} x + Ox _{, P} y,

2

=

Ox _{, p} x + Ox _{, p} y .

§ 6 Proof of (3.4)

Let C be an irreducible component of ｾ＠ d with LC

=

1,

re

along which ｾ＠ is nonreduced anywhere.

By (5.3),there are two possibilities 1m ｾ＠ ｾ＠ 0c or 0C(-l).

The case 1m ｾ＠

=

0c was discussed in section 5. In this section,

we shall discuss the case 1m ｾ＠

=

0C(-l). We note that via the

isomorphism

ｉ｣Ｏｉｾ＠

ｾ＠

0cffiOcC-1), the subsheaf 0c

=

0cffi{O} of

ｉ｣Ｏｉｾ＠

is uniquely determined. First we prove

(6.1) Lemma. Assume 1m ｾ＠ - 0C(-l). Then 1m ｾ＠ is not contained

2

in 0c (= 0Cffi{O}) C IC/IC'

Proof. Assume 1m ｾ＠

=

0c(-l) C 0c to derive a contradiction.

Let p be an arbitrary point of C. Then there are two generators

x,y of IC and two generators f,g of In

,p ｾＬｰ＠ such that x mod 12 C,p

2

(resp. y mod Ic,p) generates 0cffi{O} (resp. {O}ffiO_c(-l)) in

2

Ie IIc ' and ｾＨｦＩ＠ generates 1m ｾＬ＠ ｾＨｧＩ＠

=

0, or equivalently g

, p , P

2

E 1.9. _,p nlc _,p . Since 1m ｾ＠ is contained in 0Cffi{O}, f

=

ax mod

IC2 for some a E Ox . Thus we obtain,

,p ,p

(6.1.1) I _C,p

I

.9.,p

=

Ox _,p x + Ox _,p y,

=

Ox _,p f + Ox _,p g,

2

f

=

_{ax mod Ic ,p}

2

where a E Ox _,p ' g E In nIC .

ｾＬｰＬｰ＠

Let $

=

{U.} be a sufficiently fine covering of an open

J

neighborhood of C by Stein (or affine) open sets U .• Then by

C6.1.2) fCUj,I

C)

=

f(U. , 0X)x. J J + f(U.,OX)Y' J J

f(Uj,I.Q.)

=

f(U. , OX) f. + _fCUj,OX)gj

J J

f.

₌

<X.X. mod 12 ,

J J J C

Moreover by the choice of the generators, we may assume

(6.1.3) f.

_{= .Q.jkfk}

_{mod ICI.Q.} , _gj

_{= .Q.jkgk}

_{mod ICI.Q.} ,

J

x.

₌

x

k mod 12 , Yj

= .Q.jkYk

mod 12

.

J C C

where .Q.

jk is the cocycle LC °C (1 )

1 "I<

one

₌

E H CC,OC).

Then one checks that <Xc

:=

{<Xjl

c} is a nontrivial element

o

of H (C,OC(l». Hence <Xc has a single zero at a point

Po

of C

and i t vanishes nowhere else.

If <Xlc is nonvanishing at p in (6.1.1), then f and y

generates I

C,p , so that we may take f instead of x and can

normalize g as Bym for some B E

° X,p and for some m ｾ＠ 2 so that

the restriction of B to C is not identically zero. The integer

m is independent of the choice of p.

If <Xlc has a single zero at p in (6.1.1), then z

:=

<X

forms a regular sequence at p together with the parameters x and

y. Since f

=

zx mod IC2 _,p in C6.1.1), we may assume,by a

suitable coordinate change, that I _C,p

=

° _X,p x + Ox _,p y,

f

=

zx or zx - y s for some s ｾ＠ 2.

Therefore by taking a suitable refinement of ｾ＠ if

necessary, we may assume that

(6.1.4) U. contains

_Po

iff j

=

0,

J

C6.1.5) f(Uj,I

C)

=

feU.,OX)x. J J + fCU.,OX)Y' J J

feUj,I.Q.)

=

f(U. ,Ox)x. + feU. , Ox) g. ,

J J J J

m

(6.l.6) [(UO,I

C)

=

[CUO'OX)X O + [CUO'OX)Yo

[CU_O,I.9.)

=

_{[CUO,OX)f O + [(UO,OX)go}

f ₀

=

_zoxO or _{zOxO - Yo} s Cs ｾ＠ 2)

where xo,yo and Zo form a regular sequence everywhere on UO' go

2

E IC ' and moreover

C6.l.7) _Bj (j # 0) Crespo zO) vanishes nowhere on U

ij (:

i "# j ) (resp. on U Oj ).

:: u.

1

Now we define Bo as follows. Let x

=

x_O' y

=

yO' z

=

z00

Case O. Assume fO

=

zx. Then the second generator go of 1.9. is

normalized (mod fo) into go

=

AnCx,y)x + Bn(y,Z)yn for some n ｾ＠

2, An E [CUo,IC),B_n E [CUo'OX) ,B_n being not identically zero on

C. At a general point q of C sufficiently close to p, IC

,q

(resp. I.9.,q) is generated by x and y Crespo x and ym) by C6.1.7)

because B does not vanish at q. It follows that n ｾ＠ m. We now

define

s

Next we consider the case f

=

zx - y , s ｾ＠ 2.

Case 1- Assume s

_>

m. We can choose f.

J

¢ _{0 and f.}

=

_.9.

jkfk mod IC I .9. for any j,k.

J

s

(.9.

_o

·/z)x. mod

x

=

(f 0 +y ) I z

₌

J J

Y

=

co,x.

J J

for some c_Oj and d_Oj

the other hand,

+ d_{Oj Yj}

such that cOjlc

2 B

2CY,z)y

=

such that f .

=

J

We see

IC I .9.

.

0, d

ojic

=

.9. 0j ·

x. for

J

On

go

=

A₂Cx,y)x +

for some A2 E [(Uo,I_C)' B2 E r(UO'OX)' _{Since [CU Oj ,ICI.9.) is}

2 m+l

genera ted by x., x . y . , y . ,

J J J J we have,

2

=

B

2(dO_{J J} ·y·,Z)(do_{J J} ·Y.)

=

0

2 m+1

mod (x.,x.y.,y. )

J J J J

2

m

mod (x.,x.y.,y.)

J J J J

is divisible by (d

oJ .y.)m-2. J

Case 2.

m-2 is divisible by y

Assume s ｾ＠ m.

So we define

Then on

we see g

=

ｾ＠ A ＨｺＩｹｶｳＫｾＯｺｶ＠

o

L ｶｾ＠ By (6.1.7), we

ｶＫｾＡ［Ｚｬ＠

have

L

ａｶｾＯｚｖ＠

= 0 for k

<

m, and we define

ｶｳＫｾ］ｫ＠

B

_o

=

L

ａｶｾＯｚｖＮ＠

ｾｾｾｾｾＱＸ＠

By these definitions, we have,

m

BOYO

=

go = .Q.Ojgj

m

= .Q.OjBjYj

m m

B.y. = .Q. . . B.y.

1 1 1J J J

Therefore we have

mod ICI.Q.

mod ICI.Q.

mod IcI.Q.

mod 1C1.Q.

on U_Oj

by (6.1.3)

by (6.1.5)

(i,j ¢ 0).

for any i , j .

In Case 0 and Case 1,

Bo

is holomorphic on U

O' In Case

2, B

olc is holomorphic except at Po and meromorphic at Po' and

i t has a pole at Po of order at most V

max := max{v;

vs

+ ｾ＠

=

m,

ａｖｾ＠ # O}. Clearly v

max ｾ＠ m-2 when s

<

m. Hence

BC

U. E

ｾｽ＠

is a nontrivial element of HO(C,Oc(l-m+v ))

J max

= _{{B j}

Ic;

= {O},

which is a contradiction. Thus 1m ｾ＠ is not contained in 0C.

(6.2) Completion of the proof of (3.4). Let E

=

Coker ¢ , F

=

1m ¢. Then by (6.1), E

=

0c ' F

=

0c(-l) and we may view

ｉｃＯｉｾ＠

So we consider again the

homomorphism ¢ as,

F

Let p be an arbitrary point of C. Then there are two

generators x ,y _{of IC '}

,p and two generators f, g of 10 x,P such

2

IC (resp. y mod

,p

that ¢(f)

=

x, ¢(g)

=

0, and that x mod

generates F (resp. E). Since f

=

x mod IC 2 _,p ' we may

take f instead of

x.

Then in the same manner as in (5.3), by

taking a sufficiently fine covering ｾ＠

=

{U.} of an open

J

neighborhood of C by Stein open sets Uj , we have Xj E r(Uj,I.Q.)

'Yj E r(Uj,I_c )' B_j E _{r(uj,ox) such that Xj and Yj (resp. Xj and}

m

BjY j ) generate r(Uj,I c ) (resp. r(Uj,I.Q.))'

Since

ｉ｣Ｏｉｾ＠

=

F ffi E

=

0c(-l)ffiOc,we may assume

2 2

(6.2.1) Xj

=

.Q.jkXk mod IC' Y_j

=

_{Yk mod IC}

1

*

where .Q.

jk stands for the one cocycle Lc E H (c,oc). Since

Ker ¢ is isomorphic to 0C(-l), and i t is generated locally by

m

B.y., we see that

J J

(6.2.2)

In particular,

BC

:=

{B

j1c; Uj E ｾｽ＠ is a nontrivial

o

element of H (C,Oc(l)) by (6.2.2). Consequently

Bc

has a single

zero at a unique point PO E C and it vanishes nowhere else.

Then Z . -

.

-

BO

forms a regular sequence at

Po

with the parameters

x and y.

reducible. Let I

C' be the ideal sheaf of Ox defining C'. Then

Ie'

=

Ox

x +

Ox

z.

,Po ,Po ,Po By the assumption in (3.4), we have 6

:=

LC'

ｾ＠

O. Let

ｉ｣ＬＯｉｾＬ＠

ｾ＠

0c,(a)ffiOc,(b), a

ｾ＠

b, and

ｾｃＧ＠

:

ＨｉｾＯｉｾＩｾｏ｣Ｌ＠

ｾ＠ ｉｃＬＯｉｾＬ＠

the natural homomorphism. Then since

ｾ＠

is reduced generically along C', we have by (3.9) a + b

=

-36+2,

and dim(Coker ｾｃＧＩ＠

=

a + b + 26

=

-6 + 2. Since

BO

=

z is a

local parameter, Coker ｾｃＧ＠ p - C[y)/(ym), whence m ｾ＠ -6 + 2. , 0

Hence m

=

2, 6

=

0, Coker

ｾｃＧ＠

=

Coker

ｾｃＧ＠

,PO

=

C[y)/(y2). By

Coker ｾ＠ , we have

'l'C' , Po

a

=

2, b

=

O. Moreover this shows that C' meets no irreducible

component other than C.

§ 7 Proof of (3.5)

Assume that there is an irreducible component Co of ｾｲ･､＠

with LC_O

=

1 such that ｾ＠ is reduced at a point of Co' Assume

moreover that Co intersects an irreducible component C

1 of ｾｲ･､＠

not contained in Bs !LI.

(7.1) Lemma. Let C

=

CO' We have,

(7.1.1) C intersects the unique irreducible component C' of

ｾｲ･､＠ - C (:= the closure of ｾｲ･､｜＠ C) at a unique point p

transversally, to be more precise, we can choose local

parameters X,Y and z at p such that

(7.1.2)

along C' .

Proof.

I

C,p

=

Ox , P x + Ox , P Y,

I _ｾＬｰ＠

=

°

_X,p x + Ox _,p ZY,

ｾ＠ is reduced everywhere on C, and reduced generically

We consider the

By (3.10), ¢ is injective and Coker ¢

=

0C/Oc(-l)

=

C. Let p be

Supp Coker ¢. The in the same manner as in (5.3), we can find

local parameters x,y,w and a germ

B

E Ox such that

,p

I

where B(O,w) is not identically zero and m ｾ＠ 1. Since ¢ is

injective, we have m

=

1. Moreover we see that B(O,w) has a

single zero at p. Hence B(y,w) forms a parameter system at p

with x and y. So (7.1.1) is clear by setting z

=

B(y,w).

(7.1.2) is clear from (7.1.1). q.e.d.

(7.2) Completion of the proof of (3.5). By the assumption,

the irreducible component Co of ｾｲ･､＠ with LC

O

=

1 intersects an

irreducible component C

1 of ｾｲ･､＠ which is not contained in

Bs

ILl.

Then LC

l

=

0 or 1. Assume LCl

=

O. =

0c

(a)ffiO

c

(b), a ｾ＠ b. By (7.1.2), ｾ＠ is reduced generically

1 1

along C

1' whence the natural homomorphism ¢l

2

ICIIC is injective. Hence a ｾ＠ 0, b ｾ＠ O.

1 1

Moreover a + b

=

-3LC

1 + 2

=

2. Therefore dim Coker ¢1

=

2, (a,b)

=

(1,1) or

(2,0). Since ¢1 is not surjective at Po

:=

Co

n

C_l' there is a

unique point P1 of C

1' P1 # Po such that ¢1 is not surjective at

Pl' By the same argument as in (7.1), we can choose local

Consequently there is the third component C

2 of ｾｲ･､＠

intersecting C

1• Then C2 is not contained in Bs ILl.

Otherwise, C 1 is contained in Bs ILl because LC 1

=

o.

Hence LC

2

=

0 or 1. If LC

2

=

0, then we repeat the same argument as

above and after a finite repetition of these steps, we

eventually obtain C

m and a chain of rational curves C1' . . . 'Cm_1

of ｾｲ･､＠ such that LC

j

=

0 (1 ｾ＠ j ｾ＠ m-l) and LCm

=

1, and the

pair C

j and Ck (j

<

k) intersect at a unique point Pj

transversally iff j

=

k-l. By the same argument as above no C.

(0 ｾ＠ j ｾ＠ m) is contained in Bs ILl. Moreover by (5.1) and

=

0c ffiO

C (-1) and Cm intersects Cm-1 only.

m m

By

(2.4.4), ｾ＠ is connected so that i t is the union of CO' ... ,C

m.

Hence ｾ＠ is reduced everywhere.

Thus the proof of (3.5) is complete.

§ B Proof of (3.6)

(B.1) Lemma. Let C be an arbitrary irreducible component of

ｾ＠ _{re - -}d with LC

=

1. Then we have,

(8.1.1)

ｉｃＯｉｾ＠

ｾ＠

_{0c ffi 0C(-l),}

(8.1.2) C intersects ｾ＠ _red - C

(:=

the closure of ｾ＠ _re d' C) at a

uniaue point p transversally, to be more precise, we can choose

local parameters x,y and z at p such that

I _C,p

I

ｾＬｰ＠

=

Ox _,p x

=

Ox _,p x

+ Ox _,p y,

+ Ox Zym ,p

for some m ｾ＠ 1, we call m the multiplicity of C in ｾ＠

(B.1.3) C is not contained in Bs ILl.

Proof. The assertion (B.1.1) follows from (5.1). If ｾ＠ is

reduced at a point of C, then (B.1.2) follows in the same manner

as in (7.1). Next we consider the case where ｾ＠ is nonreduced

along C. Consider the natural homomorphism

¢ :

ＨｉｾＯｉｾＩ｀ｏ｣＠ 2

2

Ic/IC. Then by (3.3) and (6.1), 1m

¢

=

0C(-1) and 1m

¢

is not contained in 0Cffi{O}. Let E

=

Coker ¢, F

=

1m ¢. Then we may

view IC/Ic 2

=

E ffi F and consider the homomorphism

¢

as,

2 2

¢

ＨｉｾＯｉｾＩ｀ｏ｣＠ ｾ＠

FeE

ffi

F

=

IC/IC.

Then we are able to choose an open covering ｾ＠

=

{U.} of an open

J

neighborhood of C and x.,y. and

B.

satisfying (5.3.2). Here we

J J J

may assume that

2 2

Xj

=

ｾｪｫｘｫ＠ _{mod IC' Yj}

=

_{Yk mod Ic' Bjlc}

=

ｾｪｫｂｫｬｃ＠

and that x. (resp. y.) generates F (resp. E). Hence

BC

=

J J

o

and z = B

j at Po form a regular parameter system at PO. This

completes the proof of (8.1.2).

Now we are able to construct a partial "normalization" of

ｾ＠ by using the expression (8.1.2) of IC and ｉｾ＠ as follows;

With the notations in (8.1.2), we define an ideal

subsheaf ＱｾＬ＠ of Ox by;

ＱｾＬ＠ _,p = I (p E X \ C)

ｾＬｰ＠

ＱｾＬ＠ _,p = _{Ox ,p x + °X,pz (p = po)}

ｉｾＬ＠ _,p = _{° X,p} (p E C \ {po} )

where _{I ｾＧ＠} is the stalk of _ＱｾＬ＠ at p. Let 51.' be an analytic

,p

subspace of X with ｾｾ･､＠ = Ｈｾｲ･､＠ \ C) U {po}, 051.f = 0X /1 51." and

Ik =

ｉｾ＠

+ 151. (1

ｾ＠

k

ｾ＠

m). Then we have exact sequences;

(8.1.4)

o

_051.

(8.1.6) 0 _{ｾ＠ ｉｫＫＱｾＬＯＱｫ＠ ｾ＠ 0X/1k ｾ＠ 0X /1 k+ 1 51.' ｾ＠ 0}

0,

We note that

ＰｘＯＱｫＫｉｾＬ＠

_{= C[y)/(yk),1k_l/l k}

=

_{0c,Ik_lnI51.,/IknI51.'}

o

Let Vk = H ﾫｉｫＫｉｾＬＯｉｫＩｾｏｘＨｌﾻＬ＠ ni,j the natural

homomorphism of V. into V. for i

>

j.

1 J From (8.1.4)-(8.1.6)

tensored by 0x(L), we infer long exact sequences,

(8.1.7) 0

ｾ＠ ｈｏＨｏｾＨｌﾻ＠

ｾ＠ ｈｏﾫｏｸＯｉｾＬＩＨｌﾻｦｦｩｈｯﾫｏｸＯｬｭＩＨｌﾻ＠

ｾ＠

Cm

(8.1.8)

°

ｾ＠

HO(OC)

ｾ＠

V_k

ｾ＠

V_k- _l

ｾ＠

°

(8.1.9)

°

ｾ＠

V

k

ｾ＠

HO«OX/1k)(L»

ｾ＠ ＰｸＯｬｫＫｉｾＬ＠

(= C

k )

Then by (8.1.9), V_k = Ker(Ho«Ox/1k)(L»

ｾ＠ ＰｸＯＱｫＫＱｾＬＩＬ＠

whereas

°

Vm is a subspace of H ＨｏｾＨｌﾻ＠ by (8.1.7). By (8.1.8), nk,k-l

is surjective and dim V_k = dim V

k- 1 + 1, whence nm,l =

nm,rn-1 nrn-1,rn-2··

.n

2,1 Vrn ｾ＠ V1 is surjective. Since V1 =

o

0

o

nontrivial subspace of H (C,L

C)' C \ {po} is disjoint from Bs

ｉｌｾｉ＠ (= Bs ILl by (2.6». This completes the proof of (8.1.3).

q.e.d.

(8.2) Corollary.

Proof. By the above proof, dim V

k

=

dim V1 +k-1. From the exact sequence

o ｾ＠ (I

k_1/Ik)(L) ｾ＠ (Ox/1k)(L) ｾ＠ (OX/1k_1)(L) and (I

k_1/Ik)(L)

ｾ＠

0c(l), we infer hO«OX/Ik)(L))

=

o

h «OX/Ik_l)(L» + 2, whence the second assertion.

(8.3) Proof of (3.6) Start. Assume that there is an

o

q.e.d.

irreducible component C of ｾｲ･､＠ with LC

=

1 such that C

intersects an irreducible component C' of ｾｲ･､＠ not contained in

Bs ILl. Then by (3.2)-(3.5) and (3.7), Bs ILl is empty so that

for any ｾＧ＠

=

D"nD" ,D", D" E ILl, any irreducible component C"

ｯｦｾＧ＠ with LC"

=

1 intersects a component C" of ｾＧ＠ d not

red re

contained in Bs ILl. Therefore it remains to consider the case

where for any ｾ＠

=

DnD' ,D, D' E ILl, any irreducible component

C of ｾｲ･､＠ with LC

=

1 intersects a component of ｾｲ･､＠ contained

in Bs ILl. Then C is not contained in Bs ILl and there is a

unique irreducible component C' of ｾ＠ d intersecting C by (8.1). re

In what follows, we assume this to derive a contradiction in

(8.10).

First we shall prove,

(8.4) Lemma. Let C. (1 ｾ＠ j ｾ＠ s) be all the irreducible

J

components of ｾｲ･､＠ with LC_j

=

1, B. the unique irreducible

component of ｾ＠ _re d contained in Bs ILl that C. intersects. _{. J} Bv _ｾ＠

choosing a general pair D and D', BI = B2 = ••• = Bs'

Proof. We apply a variant of the argument in [11,(2.6»).

Assume the contrary. Then we can choose a one parameter family

D

t (t E p1) and a Zariski dense open subset U of p1 with the following properties;

(8.4.1) ｾｴＬｲ･､＠ = _{C1 ,t} + . . . + Cs(t),t + B1 + B2 + . . . , t E U,

B_j C Bs ILl, BI ｾ＠ B2 where ｾｴ＠

=

D

n

D

t ,

LC. t

=

I, (1 ｾ＠ j ;;; s ( t )) , J ,

(8.4.2)

(8.4.3)

U.

CI,t (resp. C₂,t) intersects BI ＨｲｾｳｰＮ＠ B₂) for any t E

Let d (resp. d

t ) be the equation defining D (resp. Dt ), and define an analytic subset Z of X x pI by Z

=

{(x,t) E X x

pI; d(x)

=

d'(x)

=

O}. Let p. be the j-th projection of X x pI,

t J

Zj all the irreducible components of Zred' gj : Yj ｾ＠ Zj the

JT. h. 1

normalization of Z., Y. ｾｊ＠ U. ｾｊ＠ P the Stein factorization of

J J J

We may assume that

= h. (u .) for some u. E U. (j = I, 2) .

J J J J

irreducible nonsingular and intersects B. only at one point

J

when v moves in a Zariski dense open subset V. of U .. Then

J J

and C

2 _{, 2} t intersect nowhere for general u1 E V1,u2 E V2.

then D' contains C _{by D}

t2C1,t₁

=

LC I t

=

1. Since Dt is

t2 1,t_{1 ' 1}

chosen general, this contradicts that C _1,t 1

is not contained in

Bs

I

L I . Therefore we may assume that C

n

C = _{CI t}

_n

_BI

=

_{C2 ,t}

2

n

_{B2 · This shows that C1,t intersects} ｾｴＬｲ･､＠ _{- C1,t at}

the intersection B1

n

B2 (# ｾＩ＠ for general t. However this is

impossible by (8.1.2). This proves that _{and C2}

t

, 2

intersect nowhere for general u

1 E V1,u2 E V2. Hence the

intersection of P1 g 1(Y1) and P1 g 2(Y2) is at most one

dimensional. However D

=

P1 g 1(Y₁)

=

P1g2CY2) because D is

irreducible. This is a contradiction. q.e.d.

By (8.4), all B. are the same, say, B. = B for any j, by

J J

choosing a sufficiently general pair D ,D' and ｾ＠ = D

n

D'. Let

-LB, and let ¢B 2 2 be the natural

n = : ( I ｾＯｉ＠ ｾＩＰｏｂ＠ ｾ＠ _IB/IB

homomorphism. One sees n ｾ＠ 0 in view of (3.2) and (8.1).

(8.5) Lemma. Let ｾ＠

=

D

n

D' for D, D' sufficiently general.

Let mj be the multiplicity of C_j in ｾＬ＠ m := m

1+ ... +ms ' Then m

=

n

+ 2

,n

ｾ＠ O.

Proof. By (8.1.2), ｾ＠ is reduced generically along B, so that ¢B

is injective by (3.10). Hence Coker ¢B is finite. One sees that

2 2

dim Coker ｾｂ＠

=

c

1CIB/IB) - ｣ＱﾫｉｾＯｉｾＩＰＰｂＩ＠

=

-LB + 2

=

n + 2, dim Coker ｾｂ＠ = m. at any intersection point PJ' := C

J.

n

B by , P_j J

C8.1.2). Since p.'s are all distinct by C8.1.2), we have

J

m ｾ＠ n + 2. Since ｾ＠ is of multiplicity mj generically along C_j

and it is reduced generically along B, ｣ｬＨｾＩ＠ is equal to

m

1c1CC1) + . . . + msclCCs) + cICB) + cICB') for some effective

S O. In fact, there is no irreducible component B" of ｂｾ･､＠ with

LBn

ｾ＠ 1 by (3.2) and (8.1) . Therefore we have by (2.2),

2 ｾ＠ L3 ::: L.Q. :;;; _L(m

1C1 + + m C s s + B) ::: m - n.

This proves m ｾ＠ n + 2, hence m ::: _{n + 2, L3} ::: _{2. q.e.d.}

(8.6) Corollary. Let.Q.::: D n D' for D, D' sufficiently

ｾｧｾ･ｾｮｾ･ｾｲｾ｡ｾｬｾＮ＠ __ ｾｔｾｨｾ･ｾ｣ｾｵｾｲｾｶｾ･＠ B intersects C. (1 ｾ＠ j ｾ＠ s) only,

J

reduced everywhere along B \ UCB

n

C

j ) j

B, Bs ILl::: Bs IL.Q.I ::: B.

C8.7) Lemma. Let D" and D" _be arbitrary members of ILl, D" >4

D" , .Q.' ::: DnnD" . _{Then .Q.'}

=

m'C' ₊

1 1

...

+ m' C' s' s' + B for some ｭｾ＠ J

and s' where ｌｃｾ＠

₌

1, m'

.

.

-

-

m' +

...

+ m' ::: _{n + 2, the}

J 1 s •

structures of .Q.' , ｃｾ＠ and B at ｃｾ＠ n B are described in C8.1).

J J

Proof. The proof of (3.7) shows that BslLI :::

¢

in the cases

(3.2)-C3.5). Since Bs ILl::: B in our case, any irreducible

component ｃｾ＠ of.Q.' with ｌｃｾ＠ ::: 1 intersects B. By the above

J red J

argument, we see .Q.' ::: miCi + ••• + ｭｾＮｃｾＬ＠ + B for some ｭｾ＠ and s'

J

where LCj ::: 1, m' :::: mi + ••• + ｭｾＮ＠ ::: n + 2. The rest is clear

from (8.1). q.e.d.

(8.8) Lemma. h (0.Q.(L»

o

::: m, and n ::: -LB

>

O.

Proof. Let.Q." be an analytic subspace of X whose ideal in Ox is

m.

defined by I.Q." ::: I.Q. +

n

ｕ､ｾｳ＠

( I ) J

C.

J

of C. in.Q.. We easily see that

J

(p E .Q.red\B),

where m.::: multiplicity

m.

0x,px + 0x,pY J

°

(p ｾ＠ U

X,p ｬｾ､Ｚﾣｳ＠

Then there is an exact sequence

(p = BnC.)

J

C. )

J

m.

(B C J -+ O.

j

Since the support of ｾＢ＠ is the disjoint union of C. (1 ｾ＠ j S s)

J

we see by (8.1.9) and C8.1) h (0.Q" (L»)

o

=

that the natural homomorphism H

o

ＨｏｾＬＬＨｌﾻ＠

2Cm

1+···+ms ) =

m.

-+ Ell C J is

j

surjective. Since

ｈｏＨｏｾｃｌＩ＠

is mapped to zero in

2m,

0 0 0

H (OBCL», we have h ＨｏｾＨｌﾻ＠

=

m, h (OB(L»

=

0 . It follows

from 0B(L)

=

0BC-n) that n

>

O. q.e.d.

and

Let h y -+ X be the blowing-up of X with B center, E

=

-1

*

_*

_*

h (B)red' N

=

h L - [E], D = h D - E, D'= h D' - E, C. = the

J

proper transform of C. (1 ｾ＠ j ｾ＠ s). Then one checks

J

C8.9) Lemma. For general D and D'

C. and (D

n

D')

=

s U

J - - - red

j=l

n

D

is isomorphic to B.

ILl,

c.

is isomorphic to

J

+ ••• + m _{s s'} C B ' · - E

.-Proof. We note that a general member of ILl is nonsingular

along B. Indeed,assume that D and D' are singular at a point p

of B, that is,

T

ｾＬｰ＠

2

ID _,p C mX _{,p ' I D, ,p}

2

=

Hom(m n _><.,p Imn _><.,p ,C)

2

C mX _,p . Let ｾ＠

=

DnD'. Then

2

=

Hom(m

X ,p lID ,p +ID, ,p +mX ,p ,C)

2

whence dim Tn _.x,p

=

3. However by (8.1.2), dim Tn _.x,p ｾ＠ 2,

which is absurd. Let q

:=

B

n

C., a :=m .. It suffices to

J J

consider the problem near q to prove (8.9). Then by (8.1.2), a

In _.x,q

=

Ox _,q X +

°

_X,q zy

I

B,q

=

Ox ,q X + Ox ,q

z,

I C., q

J

=

Ox _{, q} X + Ox _{, q} y.

We may assume without loss of generality that ID

=

Ox X,

,q ,q

I

D, ,q

=

Ox ,q (x+Zya) because general D and D' are nonsigular along B. Now i t is easy to check the assertions by a direct

computation. q.e.d.

(8.10) Completion of the proof of (3.6). Since C. is a movable

J

part of

DnD',

Bs

INI

consists of at most finitely many points,

whence NC. ｾ＠

°

for any j . Let

J

2

IB/IB = 0B (a HBOB ( b) , a ii: b, c =

a-b. Then by (3.9), a + b = c + 2b = 3n+2 and E is a rational

ruled surface L

.

By (8.9), n

>

0. Let e (resp. f ) _be a c

section (resp. a fiber) of the ruling of E _c with e2

=

c, f2

=

2

0, ef

=

1. Let e_oo be a section of LC with e_oo

=

-c, ee_oo

=

0.

Let B'

.

-

E

n

D. We see [EJ

E = -e-bf, EB' = E2l) = E2(h*L

-

E)

.

--(2n+2), E3 = c 2 3n 2, N3 L3 3h*LE2- E3 2 3n

1(IB/IB) = + = + = +

=

-

(3n+2) = 0. Consequently m

1NC1 +

...

+ m NC =

NDD'

= N 3

= 0, s s

NC.

=

0, and INI is base point free.

J

Let B'

=

pe + qf EPic E. In view of (8.9), B' is

- - 1<.

isomorphic to Band p

=

1. Since [B'J

=

[Dl

E

=

(h L-[El)E

=

e +

(b-n)f, we have q

=

b-n. If B'

=

e_oo' then q

=

-c, b

=

2n+2, a-b

= -n-2

<

0. This is a contradiction. Hence

B'

ｾ＠ e_{oo '} so that

q

ｾ＠

0, b

ｾ＠

n. Therefore hO(E,N00

E)

=

hO(Lc,e + (b-n)f)

=

n + 4.

°

° .

°