Instructions for use

Title Threefolds Homeomorphic to a Hyperquadric in P4

Author(s) Nakamura, Iku

Citation Hokkaido University Preprint Series in Mathematics, 4, 0-1-B-2

Issue Date 1987-06

DOI 10.14943/48864

Doc URL http://eprints3.math.sci.hokudai.ac.jp/900/; http://hdl.handle.net/2115/45521

Type bulletin (article)

File Information pre4.pdf

Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP

Threefolds Homeomorphic to a

Hyperquadric in p4

Iku Nakamura

Series #4. June 1987

#

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

Operators on an Infinite Dimensional Manifold

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 ~ = 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

Bibliography

0-1

§ 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,-mKX) > 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

condition on Kodaira dimension. See [12,(3.3)].

0-2

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,-KX)

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

hyperquadric 0 3 . Let L be the generator of Pic X (~ 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, ~ the scheme-theoretic complete intersection D n D' of D

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

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

linear system ILl. By studying ~ and ~ d in detail, we re

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

we are able to verify;

(0.4) Lemma. ~red 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

~ is "a double line" with ~ red irreducible nonsingular, (0.4.2)

(0.4.3)

(0.4.4)

~ is "a double line" plus a nonsingular rational curve,

~ is reduced everywhere and is the union of two

0-3

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 ~ = 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 ~ 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 ~ ZL, KX = -3L, L3 is

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

vanishing of certain cohomology groups. We also prove L3 ~ 2

o and h (X,L) ~ 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 03 . In sections

4-8, we study a scheme-theoretic complete intersection ~ = D n

D' to prove the lemmas (3.2)-(3.6),

0-4

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)

c q

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. ~

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 q-th Betti number (of X)

the q-th Chern class (of X)

the first Chern class of a vector bundle E

0-5

the homology class of an irreducible curve C

hyperquadrics in p4, see (1.1)

0-6

§ 1 Hyperquadrics in p4

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

Let xi (0 ~ i ~ 4) be the homogeneous coordinate of p4, Fv = v+l

\' 2 3 L xi ' Qv a hypersurface defined by Fv = O. The hypersurface

i=O

Q~ (v = 1,2,3) is irreducible and Q3 (:= 0;) only is

nonsingular.

The hypersurface Q~ contains a

O} and a line ~ := {x O = xl = x 2 = O}.

small open neighborhood of ~ in

(resp. U) is homotopic to q (resp. ~)

conic q

Let U

We may

and that

. - Q3 n {x3 = x 4 . - 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, H3

(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) H4(Q~,Z) ~ H3 (aU,Z) ~ HO(q,Z) ~ Z.

(1.2) Lemma. There is a Weil divisor on O~ which is not an

3 3 integral multiple of a hyperplane section H of Q1 in H4 (Q1'Z),

Proof. Let a = [aO,a1 ,a2 ] be a point of the conic q, Da = the closure of {[aO,a1,a2,x3,x4] E p4 ; x 3 ,x 4 E C}. Then by

3 (1.1.3), H = 2Da in H4 (Ql'Z), q.e.d.

1-1

0.3) Lemma. Let Q b,e a quadric surface Q~ n {x4

= O}

contained in Q~. Then H4(Q~,Z) ~ H2 (Q,Z) (~ 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 it has a Weil divisor which is not (algebraically

equivalent to) an integral multiple of a hyperplane section.

§ 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 ~ line bundle on X. Assume that H (X,OX) = 0, Pic X = ZL,

L3 > 0, KX = -3L, K(X,L) ~ 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 co~plex

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) ~ 1 and c 1c 2

= 24X(X,OX) ~ 24, X(X,mL) = X(X,Ox) + m(c~ + c 2 )L/12 + m2

c 1L2/4

+ m3L 3 /6. Assume L3 = 1 to derive a contradiction. Let c 1c 2 = 24a, a ~ 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 ~ 2. Let a = 3b + 2, b

~ o. Then X(X,2L) = 7b + (21/2), which is absurd. Consequently

L3 ~ 2 and X(X,mL) ~ (m+1)(m+2)(2m+3)/6 by c 2L = c 1c 2/3 ~ 8.

q.e.d.

2-1

(2.3) Lemma. hO(X,L) ~ 5.

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

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

(2.4) Lemma. Let D and D' be distinct members of ILl, ~ = 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 q=0,1,m>0;q=2,0~m~3;q=3,0~m~2,

Hq(D,-mLD)=O for q=0,m>0;q=1,0~m~2;q=2,m=0,1,

° 1 H (~,-L~) = 0, H (~,o~) = 0,

HO(X,OX) ~ HOCD,OD) ~ HOC~,O~) ~ C,

H3 (X,-3L) ~ H2 (D,-2LD) ~ H1(~,-L~) ~ C.

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

O~ ° [11,(1.5.1) and (1.6)].

q.e.d.

(2.5) Corollary.

(2.6) Corollary. Bs ILl = Bs ILDI = Bs IL~1 .

2-2

§ 3 A complete intersection ~ = 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, ~ := D n D' the comnlete intersection of D and D'.

Let ~red = Al + ... + As be the decomposition of ~red into

irreducible comnonents. Then

(3.1.1) each A. is a nonsingular rational curve with LA. ~ 2, ---- J J

(3.1.2) if there is an irreducible component A. with LA. = 2, 1 1

then LA. ~ 1 for j ~ i. ---- J

1 Proof. By (2.4.3), H (~,O~) = 0. 1 Hence H (A.,OA ) = ° for any

J j

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

1 1 , h (~,-L~) = 1, whence h (~d,-L~ ) ~ 1. re red

Therefore

s 1 L h (A.,-LA ) =

i=1 1 i

s L hO(A.,OA (-2+LA.» ~ 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 ~ = D n D' be the complete intersection in

(3.1). Assume that there is an irreducible component C of ~red

with LC ~ 2. Then

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

curve, isomorphic to C,

3-1

(3.3) Lemma. Let ~ = D n D' be the complete intersection in

(3.1) . Assume that there is an irreducible component C of ~ red

with LC = 1 such that ~ is nonreduced anywhere along C.

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

2 2 ..... °C(-l). I +12/12 .....

I~+IC/1c °c or li ~ C C °c , then

C3.3.1) ~ red is an irreducible nonsingular rational curve,

isomorphic to C,

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

C, the ideal sheaf 1~ (resp. Ic) is aiven by;

1~ 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~ :J 2 I C'

2 Ic/IC - °c EB 0cC-1), Ic/I~ - °c(-1),

2 1~/Ic

..... °C·

(3.4) Lemma. Let ~ = D n D' be the complete intersection in

(3.1). Assume that there is an irreducible component C of ~red

with LC = 1 such that ~ is nonreduced anywhere along C. Assume

and that if ~ d is reducible, then re ----

meets an irreducible component C' of ~ d not contained in Bs re

C

ILl. Then ~ is a double line plus a nonsingular rational curve

C'. To be more precise,

(3.4.1) ~ d is the union of C and C' with LC = 1, LC' = 0 re

, the curve C intersecting C' transversally at a uniqu~ pOint PO'

(3.4.2)

C,C') is aiven at Po Qy

3-2

I = ° x + ° Y C X,p 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' ~

is a double line alon~ C in the sense of (3.3.2), and reduced

along C',

(3.4.3)

(3.5) Lemma. Let ~ = D n D' be the complete intersection in

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

component Co of ~red with LC O = 1 and that Co intersects an

irreducible comDonent C' of ~ d not contained in Bs ILl. Then, - re

(3.5.1)

(3.5.2)

~ is reduced everywhere,

there exist another irreducible component C of ~ with m

LC = 1 and a chain of irreducible components C. of ~ with LC. = m - J - J

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

Dair Cj

and Ck (j < k) intersect iff j = k-l. ~ j = k-l, then

C. 1 and C. intersect at a unique point p. (1 ~ j ~ m) J- -- J J

transversally, to be precise,

(:= the completion of 0 0 ) = 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'Cm) (3.5.3) Ic/Ic = {

0c (1) EBOC ( 1) or 0c ( 2HBOC

(C = C1' ... 'Cm- 1 )

(3.6) Lemma Let ~ = D n D' be the complete intersection in

(3.1). ~et C be an irreducible component of ~red with LC = 1. ~

3-3

~red is reducible, then C intersects an irreducible component C'

of ~red 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 ~ = D n D' be the

complete intersection. Let ~red = Al + ••• + A be the s

decomposition into irreducible components. Then cl(~) = nlcl(A l )

+ ••• + nscl(A s ) E H2 (X,Z) for some n i > ° (see [11,(2.1)]).

Since L3 = L~ = n l LA 1 + + nsLAs' there is at least a

component A. with LA. > 0. 1 1

We see that there are only three

cases;

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

Case 2. ~red contains no irreducible components C' with

LC' ~ 2, but contains an irreducible component C with LC = 1

along which ~ is nonreduced anywhere,

Case 3. ~red contains no irreducible components C' with

LC' ~ 2, but contains an irreducible component Co with LC O = 1

such that ~ is reduced at a point of CO'

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

Bs IL~I. Since L3 = L~ = LC = 2, we have L~ = O~(2), so that IL~I

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

hO(X,L) = 2 + hO(~,L~) = 5.

3-4

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

and (3.4), ~red is isomorphic to C and Ic/I~ ~ 0C(-l). Hence we

have an exact sequence,

° ~ whence ° follows that

° is exact. Hence ILl is base point free.

0,

° is exact.

HOCC,OCCl»

H1

(C,OC(1») ~ ° Moreover hOCX,L) =

The intersection number L3 = L~ = 2 because

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

complete.

It

2 +

Next we consider the case where ~red is reducible. Then

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

rational curve C' ,whence cIC~) = 2cICC) + cICC') and L~ = 2. We

2 12 = 0CC-l)+I C via the define a subsheaf 12 of IC by

isomophism Ic/I~ ~ 0CffiOcC-1). Let p = C n c' . We note that with

the notations in (3.4), 1 2 ,p (:= the stalk of 12 at p) = 0x,px +

2 Ox y . ,p

° ~ Then we have exact sequences;

° ~ ~ °c,ffiCOX/I 2 )CL)

0C(l) ~ (OX /I 2)(L)

0,

because IC/I2 = 0C. We see ° that a subspace H COC,) ffi

° 0 H (CI C/I 2 )CL» of H COC,) ffi HO(COx/I

2)CL» is mapped onto

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

° h (~,L~) + 2 = 5, Bs ILl = Bs IL~I = ~. proof of C3.7) in Case 2.

3-5

Therefore hOCX,L) =

This completes the

Case 3. By (3.5) and (3.6), ~ is reduced everywhere and ~ = Co

+ ••• + Cm with LC O = LCm = 1, LC j = ° (1 ~ j ~ m-1). Then L3 = L~ = L(C

O + ... + C )

m = 2. Consider an exact sequence,

° -+ O~(L) -+ 0c (l)®OC ffi ... ffiOc ffiOc (1) -+ em -+ 0. ° 1 m-1 m

It follows from this that hO(X,L) 2 ° 5, and = + h (~,L~) = 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~ (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 )I(

f 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). 3 Hence f(X) # 01

,

o~ 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) =

03 . Since~)I( is an isomorphism of Pic X onto Pic 03 (= Z[H),

the exceptional set (det(Jac f» of f is empty (see [11,(2.8)]).

Therefore f is an isomorphism of X onto 03 . q.e.d.

3-6

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 ~

be the natural homomorphism induced from

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

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

Proof. We note that (I.Q./I~)00c - 0c(-L)ffiOc(-L) is locally free,

hence torsion free. Therefore the following conditions are

equivalent to each other;

(3.10.1) ~ is injective everywhere,

(3.10.2) ~ is injective at a pOint q of C,

(3.10.3) Coker(~) = 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.

Thus the assertion is clear. q.e.d.

3-7

(3.11) Lemma. Let I and I' (~ Ox) be ideal sheaves of OX'

SUPpose that I C It and h1(OX/I) = 0, dim supp(Ox/I) ~ 1. Then

h1(OX/I') = 0 and X(Ox/I') ~ 1.

Clear.

3-8

§ 4 Proof of (3.2)

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

Assume that C is an irreducible component of ~ with LC red

~ 2. Then by (3.1.1), we have LC = 2. Then (I~/I~)00c -

°C(-2)ffiOC(:-2). 1 Since C = P , by a theorem of Grothendieck, we

2 express Ic/IC =

(4.1) Lemma. I~ rt 12 C

Proof. Suppose r~ C 2 h 1 (O 112) by (2.4.3). Ic· Hence = 0

X C

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

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. ~ is injective.

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

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

Grothendieck, we express Ker ~ = 0CCc), 1m ~ = 0CCd) for some

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

o.

Hence c + d = -4, b ~ c ~ -2 ~ d ~ 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 0CCd) is contained in a direct summand 0CCa) of IC/IC' Here we

note that if b < d, then b < a so that the subsheaf 0cCa) in the

4-1

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.' I/I~ :: 0C(a), Ic/I :: 0C(b). By H1

(05l.) = H1

(OX/I 5l.) =

0, we have,

1 ~ X(Ox /I ) = X(OX/lc) + XOc/I) = 2 + b,

whence b ~ -l. This contradicts b ~ c ~ -2. Hence a = b = c = d = -2. Next we let J = 1m ct> 12 15l. + = C

+ 2 I C ' Then J :J

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

1 ~ 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 ~ a, -2 ~ b, whence a = b = -2. Hence

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

isomorphic to C.

This completes the proof of (3.2).

4-2

q.e.d.

§ 5 Proof of C3.3)

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

~red with LC = 1. Then Ic/I~ - 0c ffi 0cC-1).

1 2 Proof. Since C = P , we express Ic/Ic = 0cCa)ffiOcCb), a ~ b.

Then by C3.9), a + b

We shall show a = 0, b = -1. We assume b ~ -2 to derive a

contradiction. Consider the natural homomorphism cI> . .

2 (I ~/I ~H90C ~

2 IC/lc' When b ~ -2, 1m cI> is contained in °c Ca )

= °cCa)ffi{O} . Let I = OcCa)+I~. Then IC :::J I :::J I~ and IC /I -0CCb). Hence by C3.11),

1 ~ 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 ~ d with LC = 1 along which ~ is nonreduced re

anywhere.

2 C5.2) Lemma. I~ rz. IC'

2 Proof. Assume I~ C I C '

I and I~/I = CI~/I~)~OC hOCOx/I) = 1, h 1CO x /I)

I + 14/14 ~ C C

Let I = ICI~. 2 3 Then I~ :::J I :::J I~ , IC :::J

= °c C- 1 )ffiOc 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 ~ IcI~+IC/IC = I+IC/IC (C IC/IC)

and = °C(-1)ffiOc (-1)ffiOC (-2)ffiOc (-2).

5-1

Let I' = 0CffiOC(-1)ffiOC(-2)+I~. Th n 13 ~ I' ~ I, 1 311' ~ e C C

1 0C(-3). By h (OX/I) = 0 and (3.11), we have,

1

which is a contradiction.

= X(Ox/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.)00C ~ IC/lc' By (5.2),

1m ¢ is not zero. Since 1m ¢ (~ I +12/12) is a subsheaf of a .Q. C C

torsion free sheaf IC/IC2, l·t l'S 1 110f oca y C- ree. Since .Q. is

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

set 1m ~ = 0c(c) for some c E Z. Then c = 0 or -1 because

2 (I .Q. I I .Q. ) 00 C

,." 0c(-1)ffiOc (-1). In view of our assumption in

(3.3),

= 1m ¢

= 0, EV

1m ¢ ~

~ °C·

being

0C' Ker ¢ ~ 0C(-2). Let E = Coker ¢ ~ 0C(-1), F

Then we may view Ic/I~ = E ffi F because H1 (C,Ev0F)

the dual of E. 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 ' 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 mod 2 it is to = x IC ' easy see ,p

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

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 EO, m > 0 such that ,p X,p

(5.3.1) I C,p = Ox x + Ox y , p , p

5-2

m I~,p = 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 ~ 2 because

2 g E IC . ,p

Let ~ = {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.,I A ) ,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

r(Uj,I~) r(u., 0X)x. m

= + r (U . , Ox ) B . y .. J J J J J

Since 2 F EB E "-

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

(5.3.3) mod 12 ~jkYk mod 2 x. = x k

, y. = IC' J C J 1 *­

where ~jk 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 ¢ (~ 0C(-2» are

identified iff (we may assume that)

(5.3.4)

This shows that

( 5 . 3 . 5 )

In particular, Bc := {Bj1c ; Uj E ~} 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 ~ d is nonsingular re

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

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

any point p of C, there exist local parameters x, y E IC such ,p

that

5-3

I C,p

I .Q.,p

= Ox x + Ox y, , p , P 2 = Ox x + Ox y . , p , p

This completes the proof of (3.3).

5-4

q.e.d.

§ 6 Proof of (3.4)

Let C be an irreducible component of ~ d with LC = 1, re

along which ~ is nonreduced anywhere.

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

The case 1m ~ = 0c was discussed in section 5. In this section,

we shall discuss the case 1m ~ = 0C(-l). We note that via the

isomorphism Ic/I~ ~ 0cffiOcC-1), the subsheaf 0c = 0cffi{O} of

Ic/I~ is uniquely determined. First we prove

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

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

Proof. Assume 1m ~ = 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 ~,p such that x mod 12 C,p

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

2 Ie IIc ' and ~(f) generates 1m ~, ~(g) = 0, or equivalently g , p , P

2 E 1.9. nlc . ,p ,p Since 1m ~ 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 x + Ox y, ,p ,p

= Ox f + Ox g, ,p ,p 2

f = ax mod Ic ,p 2

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

Let $ = {U.} be a sufficiently fine covering of an open J

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

6-1

C6.1.2) fCUj,I C ) = f(U. , 0X)x. + f(U.,OX)Y' J J 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 it 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 ~ 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 in C6.1.1), we may assume,by a ,p

suitable coordinate change, that I = ° x + Ox y, C,p X,p ,p

f = zx or s zx - y for some s ~ 2.

Therefore by taking a suitable refinement of ~ 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. + fCU.,OX)Y' J J J J

feUj,I.Q.) = f(U. ,Ox)x. + feU. , Ox) g. , J J J J

m for j # 0, gj = Bjyj' Bj E feUj,Ox)' m ~ 2, Bj1c being not

identically zero,

6-2

(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 s Cs ~ 2) zOxO - Yo

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 Uij (:

i "# j) (resp. on UOj ).

:: 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 ~

2, An E [CUo,IC),B n E [CUo'OX) ,Bn 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 ~ m. We now

define

s Next we consider the case f = zx - y , s ~ 2.

Case 1- Assume s > m. We can choose f. J

¢ 0 and f. = .9. jkf k 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 B2

CY,z)y

=

such that f . = J

We see

IC I .9. .

0, dojic = .9. 0j ·

x. for J

On

go = A2 Cx,y)x +

for some A2 E [(Uo,IC)' 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

2 g() = B2 (y,z)y

we have,

6-3

j

2 = B2 (d O·y·,Z)(do ·Y.) J J J J

= 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 (do

.y.)m-2. J J

Case 2.

m-2 is divisible by y

Assume s ~ m.

So we define

Then on

we see g = ~ A (z)yvs+~/zv o L v~ By (6.1.7), we v+~!;:l

have L Av~/ZV = 0 for k < m, and we define vs+~=k

B = o L Av~/ZV. ~~~~~18

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 UOj

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 UO' In Case

2, Bolc is holomorphic except at Po and meromorphic at Po' and

it has a pole at Po of order at most Vmax := max{v; vs + ~ = m,

AV~ # O}. Clearly vmax ~ m-2 when s < m. Hence BC

U. E ~} is a nontrivial element of HO(C,Oc(l-m+v )) J max

= {Bj Ic;

= {O},

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

q.e.d.

6-4

(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 IC/I~

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 such x,P

2 IC (resp. y mod ,p that ¢(f) = x, ¢(g) = 0, and that x mod

generates F (resp. E). Since f 2 = x mod IC ' we may ,p

take f instead of x. Then in the same manner as in (5.3), by

taking a sufficiently fine covering ~ = {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 )' Bj 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 Ic/I~ = F ffi E = 0c(-l)ffiOc,we may assume

2 2 (6.2.1) Xj = .Q.jkXk mod IC' Yj = 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 it is generated locally by

m B.y., we see that J J

(6.2.2)

In particular, BC := {Bj1c

; Uj E ~} 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.

The curve C intersects a unique irreducible component C'

of .Q.red at PO' but nowhere else. In particular, .Q.red is

6-5

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' ~ O. Let Ic,/I~, ~ 0c,(a)ffiOc,(b), a ~ b, and ~C' :

(I~/I~)~Oc, ~ IC,/I~, the natural homomorphism. Then since ~

is reduced generically along C', we have by (3.9) a + b = -36+2,

and dim(Coker ~C') = a + b + 26 = -6 + 2. Since BO = z is a

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

Hence m = 2, 6 = 0, Coker ~C' = Coker ~C' ,PO = C[y)/(y2). By

Coker ~ , we have 'l'C' , Po

a = 2, b = O. Moreover this shows that C' meets no irreducible

component other than C.

Thus the proof of (3.4) is complete. q.e.d.

6-6

§ 7 Proof of (3.5)

Assume that there is an irreducible component Co of ~red

with LC O = 1 such that ~ is reduced at a point of Co' Assume

moreover that Co intersects an irreducible component C1

of ~red

not contained in Bs !LI.

(7.1) Lemma. Let C = CO' We have,

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

~red - C (:= the closure of ~red\ 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 x + Ox Y, , P , P

I = ° x + Ox ZY, ~,p X,p ,p

~ 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 C,p = Ox x + Ox Y, ,p ,p

7-1

where B(O,w) is not identically zero and m ~ 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 ~red with LC O = 1 intersects an

irreducible component C1 of ~red which is not contained in

Bs ILl. Then LCl

= 0 or 1. Assume LC l = O. =

0c (a)ffiOc (b), a ~ b. By (7.1.2), ~ is reduced generically 1 1

along C 1 ' whence the natural homomorphism ¢l

2 ICIIC is injective. Hence a ~ 0, b ~ O.

1 1 Moreover a + b =

-3LC1

+ 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 C1' P 1 # Po such that ¢1 is not surjective at

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

parameters x,y,z at P1 such that

7-2

Consequently there is the third component C2 of ~red

intersecting C1

• Then C2

is not contained in Bs ILl.

Otherwise, C1 is contained in Bs ILl because LC 1 = o. Hence LC2

= 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 Cm and a chain of rational curves C1

' ... 'Cm

_1

of ~red such that LC j = 0 (1 ~ j ~ m-l) and LCm = 1, and the

pair Cj

and Ck (j < k) intersect at a unique point Pj

transversally iff j = k-l. By the same argument as above no C.

(0 ~ j ~ 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), ~ is connected so that it is the union of CO' ... ,Cm.

Hence ~ is reduced everywhere.

Thus the proof of (3.5) is complete.

7-3

J

§ B Proof of (3.6)

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

~ d with LC = 1. Then we have, re --

(8.1.1) IC/I~ ~ 0c ffi 0C(-l),

(8.1.2) C intersects ~ - C (:= the closure of ~ d' C) at a red re

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 ~,p

= Ox x ,p

= Ox x ,p

+ Ox y, ,p

+ Ox Zym ,p

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

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

Proof. The assertion (B.1.1) follows from (5.1). If ~ 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 ~ is nonreduced

along C. 2 Consider the natural homomorphism ¢ : (I~/I~)@Oc

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 2 IC/Ic = E ffi F and consider the homomorphism ¢ as,

2 2 ¢ (I~/I~)@Oc ~ FeE ffi F = IC/IC.

Then we are able to choose an open covering ~ = {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 = ~jkXk mod IC' Yj = Yk mod Ic' Bjlc = ~jkBklC

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

o {BjIC;Uj E ~} is a nontrivial element of H (C,OC(1», whence BC

has a single zero at a unique point PO of C. Then x = X.,y = y. J J

B-1

and z = Bj

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

~ by using the expression (8.1.2) of IC and I~ as follows;

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

subsheaf 1~, of Ox by;

1~, = I (p E X \ C) ,p ~,p

1~, = Ox x + °X,pz (p = po) ,p ,p

I~, = ° (p E C \ {po} ) ,p X,p

where I ~' is the stalk of 1~, at p. Let 51.' be an analytic ,p

subspace of X with ~~ed = (~red \ C) U {po}, 051.f = 0X /1 51." and

Ik = I~ + 151. (1 ~ k ~ m). Then we have exact sequences;

(8.1.4) o 051.

(8.1.6) 0 ~ Ik+1~,/1k ~ 0X/1k ~ 0X /1 k+ 151.' ~ 0

0,

We note that 0X/1k+I~, = C[y)/(yk),1k_l/l k = 0c,Ik_lnI51.,/IknI51.'

o Let Vk = H «Ik+I~,/Ik)~OX(L», 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 ~ HO(O~(L» ~ HO«Ox/I~,)(L»ffiHo«Ox/lm)(L» ~ Cm

(8.1.8) ° ~ HO(OC) ~ Vk ~ Vk - l ~ ° (8.1.9) ° ~ Vk ~ HO«OX/1k)(L» ~ 0x/lk+I~, (= C

k)

Then by (8.1.9), Vk = Ker(Ho«Ox/1k)(L» ~ 0x/1k+1~,), whereas

° Vm is a subspace of H (O~(L» by (8.1.7). By (8.1.8), nk,k-l

is surjective and dim Vk = dim Vk - 1 + 1, whence nm,l =

nm,rn-1 nrn-1,rn-2·· .n2 ,1 Vrn ~ V1 is surjective. Since V1 =

o 0 H (C,LC-PO)(:= elements of H (C,LC ) vanishing at PO) is a

8-2

o nontrivial subspace of H (C,LC)' C \ {po} is disjoint from Bs

IL~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 Vk

= dim V1

+k-1. From the

exact sequence

o ~ (I k _1/I k )(L) ~ (Ox/1k)(L) ~ (OX/1k_1)(L)

and (I k _ 1 /I k )(L) ~ 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 ~red with LC = 1 such that C

intersects an irreducible component C' of ~red not contained in

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

for any ~' = D"nD" ,D", D" E ILl, any irreducible component C"

of~' with LC" = 1 intersects a component C" of ~' d not red re

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

where for any ~ = DnD' ,D, D' E ILl, any irreducible component

C of ~red with LC = 1 intersects a component of ~red contained

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

unique irreducible component C' of ~ 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 ~ j ~ s) be all the irreducible J

components of ~red with LC j = 1, B. the unique irreducible

J

8-3

component of ~ d contained in Bs ILl that C. intersects. Bv re . J ~

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) ~t,red = C1,t + ... + Cs(t),t + B1 + B2 + ... , t E U,

Bj C Bs ILl, BI ~ B2 where ~t = D n Dt ,

LC. t = I, (1 ~ j ;;; s ( t )) , J ,

(8.4.2)

(8.4.3)

U.

CI,t (resp. C2 ,t) intersects BI (r~sp. 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 ~ Zj the

JT. h. 1 normalization of Z., Y. ~J U. ~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 C2 t intersect nowhere for general u 1 E V1,u 2 E V2 . , 2

then D' contains C by Dt2

C1,t1 = LC I t = 1. Since Dt is

t2 1,t1 ' 1

chosen general, this contradicts that C 1,t1 is not contained in

Bs I L I . Therefore we may assume that C n C = CI t n BI 1,t l 2,t2 ' I

8-4

= C2,t2n B2 · This shows that C1,t intersects ~t,red - C1,t at

the intersection B1 n B2 (# ~) for general t. However this is

impossible by (8.1.2). This proves that and C2 t , 2

intersect nowhere for general u1

E V1,u 2 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(Y 1 ) = 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 ~ = D n D'. Let

-LB, and let ¢B 2 2 be the natural n = : (I ~/I ~)0OB ~ IB/IB

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

(8.5) Lemma. Let ~ = D n D' for D, D' sufficiently general.

Let mj be the multiplicity of Cj in ~, m := m1+ ... +m s ' Then m

= n + 2 ,n ~ O.

Proof. By (8.1.2), ~ 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 ~B = c 1 CI B/I B) - c1«I~/I~)00B) = -LB + 2 = n + 2, dim

Coker ~B = m. at any intersection point PJ' := CJ. n B by

, P j J

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

m ~ n + 2. Since ~ is of multiplicity mj generically along Cj

and it is reduced generically along B, cl(~) is equal to

m1 c1CC 1 ) + ... + msclCC s ) + cICB) + cICB') for some effective

one cycle B' such that Supp B' C Bs ILl, Supp B' ~ B. Then LB'

8-5

S O. In fact, there is no irreducible component B" of B~ed with

LB n ~ 1 by (3.2) and (8.1) . Therefore we have by (2.2),

2 ~ L3 ::: L.Q. :;;; L(m1

C1

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

This proves m ~ n + 2, hence m ::: n + 2, L3 ::: 2. q.e.d.

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

~g~e~n~e~r~a~l~. __ ~T~h~e~c~u~r~v~e B intersects C. (1 ~ j ~ s) only, J

reduced everywhere along B \ UCB n Cj

) 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 m~ J

and s' where LC~ = 1, m' . - m' + + m' ::: n + 2, the . - ... J 1 s •

structures of .Q.' C~ and B at C~ 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~ of.Q.' with LC~ ::: 1 intersects B. By the above J red J

argument, we see .Q.' ::: miCi + ••• + m~.C~, + B for some m~ and s' J

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

from (8.1). q.e.d.

(8.8) Lemma. o h (0.Q.(L» ::: 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

Ud~s

(I ) J C.

J

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

(p E .Q.red\B),

8-6

where m.::: multiplicity J

m. 0x,px + 0x,pY J

° (p ~ U X,p l~d:£s

Then there is an exact sequence

(p = BnC.) J

C. ) J

m. (B C J -+ O. j

Since the support of ~" is the disjoint union of C. (1 ~ j S s) J

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

that the natural homomorphism o H (O~,,(L»

2Cm 1 +···+m s ) =

m. -+ Ell C J is

j

surjective. Since HO(O~CL) is mapped to zero in

2m,

000 H (OBCL», we have h (O~(L» = 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 ~ j ~ 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 C B '·- E s s' .-

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 ~,p

2 ID C mX ' I D, ,p ,p ,p

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

2 C mX . ,p Let ~ = DnD'. Then

2 = Hom(mX lID +ID

, +mX

,C) ,p ,p ,p ,p 2 = HomCmx Imx ,C) ,p ,p

8-7

whence dim Tn = 3. However by (8.1.2), dim Tn ~ 2, .x,p .x,p

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 = Ox X + ° zy .x,q ,q X,q

I B,q = Ox X + Ox z, ,q ,q I C., q

J = Ox X + Ox y. , q , q

We may assume without loss of generality that ID = Ox X, ,q ,q

I D, = Ox (x+Zya) because general D and D' are nonsigular ,q ,q

along B. Now it 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. ~ ° for j . Let any 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 with e 2 = c, f2 = c 2

0, ef = 1. Let e oo be a section of LC with e oo = -c, eeoo = 0.

Let B' . - E n D. We see [EJ E = -e-bf, EB' = E2l) = E2(h*L - E) . --(2n+2), E3 2 3n 2, N3 L3 3h*LE2- E3 2 3n = c

1(I

B/I

B) = + = + = +

=

- (3n+2) = 0. Consequently m1NC 1 + ... + m NC = NDD' = N3 = 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' ~ e oo ' so that

q ~ 0, b ~ n. Therefore hO(E,N00E

) = hO(Lc,e + (b-n)f) = n + 4.

° ° . ° We have by (8.8) h (Y,N) = h (X,L) = h (~,L~) + 2 = m + 2 = n +

8-8

4, and by (8.1.2) hO(Y,N-E) = hO(X,I~L) = 0, whence we have a

natural isomorphism HO(Y,N) - HO(E,N00E

).

n+3 . Let g : Y ~ P (n > 0) be the morph1sm associated with

the linear system INI, gE the restriction of g to E. Any point

y of Y is contained in some C. of some D n D' because hO(Y,N) ~ J

5. By NC. = 0, J

C. is mapped to one point. J

Moreover EC. = J

- )I( -(N+E)C. = h LC.

J J = LC. = 1, whence g(y) = g(a.nE). Hence g(Y) =

J J

g (E) • We note that the linear system IN00E I defines an

isomorphism gE of E into pn+3 iff b > n. If b = n, then gE is

an isomorphism over E \ e oo '

We shall define a morphism ~ of Y onto E as follows. If

b > n, then we define ~ to be the morphism g. In the general

case, take an arbitrary point y of Y. Then choose two general

members 0 and 0' of INI such that D,D' pass through y. The

intersection ~ d = (D n D') d is the disjoint union of C. (j = re re J

1, ... ,s) by (8.9).

y. Since EC. = 1, E J

We define ~(y) = y'

the choice of D and

such that Oil and Oil

Hence there is a unique C. passing through J

intersects C. at a unique point y' of E. J

= E n a .. The point y' is independent of J

0' . To show this, take Oil and Oil of INI

pass through y. Let 9.' = Oil n Oil , 9.' red

a 1 +···+CS

" Then there is a unique ai passing through y. In

- -fact, since DC! = D'C! 1 1

= NC! = 0, both D and D' contain C!, 1 1

whence Ci is also the unique irreducible component of ~red

passing through y. Hence C. - C' J - i' Therefore c. n E = C! n E. J 1

=

It is easy to check that for any point y of E (= Ec )' there

exist two members Hand H' of IN00E

I such that H n H' is reduced

and it contains y. By the isomorphism HOCY,N) = HOCE,N00E

), for

8-9

a gi ven y of Y, we can choose two members D" and D'" of I L I such

that D" n D m n E is reduced and D" , D'" pass through y where D" *- *-

h D" - E, D'" = h D'It - E. Hence D" n D'" has m~ = 1 for any j in J

(8.7) . By using this it is easy to see that tV is a morphism of

into (indeed, onto) E. If b = n, the morphism tV coincides with

-1 the morphism g on Y\g (g(e oo )).

E\e oo •

One also sees readily that any fiber of tV endowed with

reduced structure is p1, one of the irreducible components of

D n D' for some D, D' E INI. Since both Y and E are smooth, tV

is flat and any general (scheme-theoretic) fiber of tV is pl.

Any fiber is mutually algebraically equivalent and tV-1 (X)E = 1

=

Y

for general x E E. By the criterion of multiplicity one, we see

that a fiber tV- 1 (x) is generically reduced for any x E E. Since

any fiber is Gorenstein, it is therefore reduced everywhere,

whence it is a nonsingular rational curve pl. Thus the morphism

tV gives a p1-bundle structure of Y over E with a section E.

Therefore there is a rank two vector bundle F on E such that Y ~

P(F) and the following is exact;

o ~ 0E ~ F ~ det F ~ o.

The surface E (= P(det F)) is embedded into Y by viewing det F

as a quotient bundle of F as above. Then we have det F = -NE/y

-[EJ bf. Since 1 0, F '" det F EB °E· Now = = e + H (E,-e-bf) = E it is easy to see that b

2(Y) (or Picard number of Y) = 3. The

threefold X is obtained from Y by contracting E to a curve B,

whence b2

(X) (or Picard number of X) = 2. This is a

contradiction.

8-10

This completes the proof of (3.6).

(8.11) Remark. The proofs in sections 2-8 ~ork as well in

arbitrary characteristic.

8-11

§ 9 Proof of (0.1)

(9.1) Theorem. Let X be a compact complex threefold

homeomorphic to p3 is isomorphic to p3 if H1 (X,OX) = ° and if

hO(X,-mKx) ~ 2 for some positive integer m.

1 Proof. Since H (X,OX) = 0, we have a natural exact sequence

1 ~ H1(X,O~) ~ H2 (X,Z)(= Z) ~ H2 (X,OX)' Since H1(X,O~)

has a nontrivial element KX and H2 (X,OX) is torsion free,

1 * 2 H (X,OX) is mapped isomorphically onto H (X,Z). Let L be the

1)1( 3 generator of H (X,OX) with L = 1. Since the second Stiefel

Whitney class w2 (= c 1 (X) mod 2) and rational Pontrjagin classes

are topological invariants, we see by [S,pp. 207-208]

c 1 (X) = (2s+4)L, X(X,OX) = (s+1)(s+2)(s+3)/6

for integer We 3 hO(X,Q~) :it b3 ° X(X,OX) an s . see h (X,OX) = = ,

1 2 ~ 1. This shows = + h (X,OX) s ~ 0, KX = -(2s+4)L, 2s+4 ~ 4.

Thus all the assumptions in [11,(1.1)] are satisfied. Hence X

., h' t p3 1S lsomorp lC 0 • q.e.d.

(9.2) Proof of (0.1). By the same argument as in (9.1), we see

Pic X ~ H2 (X,Z) = Z. Let L be a generator of Pic X with L3 = 2.

Then by [1, p.188] or [10, p.321], we have,

c1

(X) = (2s+3)L, X(X,OX) = (s+1)(s+2)(2s+3)/6

for an integer s. By (1.2), X(X,Ox) ~ 1, whence s ~ O.

If s > 0, then X is isomorphic to p3 by [11,(1.1)] which is a

contradiction. Hence s = ° and X is isomorphic to Q3 by (2.1).

q.e.d.

Theorems (0.2) and (0.3) are derived from (0.1). See

[11],[12),

9-1

Appendix.

As was pointed out by Fujita, the proof of C2.7) in [11)

does not work in positive characteristic because it uses

Bertini's theorem.

We shall give an alternative proof of [11,(1.1)) in

arbitrary characteristic.

CA.1) Theorem. Let X be a complete irreducible nonsingular

algebraic threefold defined over an alaebraically closed field of

arbitrary characteristic. Assume Pic X = ZL, H1 (X,OX) = 0, KX =

-dL Cd ~ 4), L3 > 0, KCX,L) ~ 1. Then X is isomorphic to p3.

For the proof of CA.1), in view of [11,C2.8) or (2.9)),it

suffices to prove that a complete intersection ~ = D n D' for any

pair D and D' E ILl is a nonsingular rational curve.

In view of [11,(2.2)), d = 4, KX = -4L. And by

[11,(2.3)), there is a unique irreducible component A1 of ~red

with LA1 = 1. Here we write C = A1 for simplicity.

First we show,

CA.2) Lemma.

Proof. Let I~ Crespo IC) be the ideal sheaf in Ox defining ~

Crespo C). By Grothendieck's theorem, let IC/1~ = 0cCa)ffiOCCb),

a ~ b. We infer c1C1C/1~)+c1CQ~) = c 1 (Qi@Oc) = KXC = -4LC = -4.

2 Hence a + b = -2. Hence b ~ -1. Assume I~ C IC' We consider

the natural homomorphism ¢ C1~/I~)00c ~ I~/I~ C=

0c(2a)ffiOc Ca+b)ffiOCC2b)). Since CI~/I~)00c - 0CC-1)ffiOCC-1), 1m ¢ 3

is contained in 0C(2a). Let I = 0CC2a)+1C ' Then since by

A-1

[11,(1.7.3») It follows

that

which is absurd. q.e.d.

CA.3) Lemma.

q, : (I.Q I I ~ ) 00 e

Ie/I~ = 0C(-1)ffiOC C-1) and the natural homomorphism

2 ~ Ie/le is an isomorphism.

Proof. By the proof of (A.2), we note a + b = -2, b ~ -1 ~ a.

If a > b, then 1m q, is contained in CeCa).

Then I.Q C I C IC' IC/I = 0eCb). In the same manner as in (A.2),

by [11,(1.7.3») and (3.11), we have

1 ~ X(OX/I) = X(OX/IC) + X(IC/I) = 2 + b.

This is a contradiction. Hence a = b = -1. Assume q, is not

injective. We note that by (A.2), q, is a nontrivial

homomorphism. Since 1m q, (= I.Q+I~/I~) is a rank one subsheaf of

2 a torsion free sheaf IC/IC' it is locally 0e-free. Here we may

set 1m q, = 0c(c) for some c E Z. Then c = -1 because (I.Q/I~)~Oe

Let E = Coker q, = 0CC-1), F = 1m q, = 0e(-1).

2 1 v v Then we may view IC/IC = E ffi F because H (C,E 0F) = 0, E being

the dual of E. So we consider again the homomorphism q, 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 I .Q,p such

that cl>(f) = x, cl>(g) = 0, x mod r2 (resp. y mod r2 ) C,p e,p

generates F (resp. E) . In the same manner as in (5.3), we

obtain local parameters x and y E Ie and B E Ox ' m > 0 such ,p ,p

that

A-2

(A.3.!) I C,p

I ,Q,p

= Ox x + Ox y , p , p

= Ox x + Ox ~ym ,p ,p

where the restriction Bc of B to

note that m ~ 2 because g E I2 C,p

C is not identically zero. We

Let cp = {U . } be a J

sufficiently fine covering of an open

neighborhood of C by Stein open sets U .. Then by CA.3.1), we J

have x. E r(U.,I o ) ,y. E rCU.,I c ), B. E rcu.,oX) such that x. and J J x J J J J J

m Yj (resp. Xj and BjY j ) generate rCUj,I c ) Crespo rCUj,I,Q))'

2 Since (I~/I,Q)~OC = 0cC-1)ffiOcC-1), we may assume that

2 2 CA.3.2) Xj = ,QjkXk mod IC' Yj = ~jkYk mod IC'

1 )I(

where ~jk stands for the one cocycle LC E H CC,Oc).

Two elements DjBjY~ and DkBkY~ E Ker ¢ (= 0cC-1)) are

identified iff (we may assume that)

(A.3.3)

This shows that

CA.3.4)

In particular, BC := {BjIC

; Uj E cp} is a nontrivial

o element of H (C,OCC1-m)). This is possible only when m = 1 and

Bc is a nonzero constant. This contradicts m ~ 2. Consequently ¢

is injective whence it is an isomorphism. This completes the

proof of (A.3). g.e.d.

By (A.3), 10 + Ic2 = IC whence 10 = IC for any x,p ,p,p x,p,p

pOint p of C. This shows that ~ is nonsingular anywhere along

C. Since ~ is connected by [11,(1.7)), ~ is isomorphic to C.

Then it is easy to see that Bs ILl = ¢ and that the morphism

associated with ILl is an isomorphism of X onto p3. q.e.d.

A-3

Bibliography.

[1] Brieskorn, E. Ein Satz tiber die komplexen Quadriken,

Math. Ann. 155 (1964), 184-193

[ 2] Fuj ita, T. : On the structure of polarized varieties with

~-genera zero, J. Fac. Sci. Univ. of Tokyo, 22 (1975), 103-115

[3) Fulton, W. : Intersection TheorY, Springer Verlag, 1984

[4) Hirzebruch, F. : Topological Methods in Algebraic Geometry,

Springer Verlag, 1966

[5] Hirzebruch, F. and Kodaira, K. : On the complex projective

spaces, J. Math. Pure Appl., 36 (1957), 201-216

[6] Iitaka, S. On D-dimensions of algebraic varieties, J.

Math. Soc. Japan, 23 (1971), 356-373

[ 7 ] Algebraic Geometry, Graduate Texts in Math., 76,

Springer Verlag, 1981

[8] Kobayashi, S., Ochiai, T. : Characterizations of complex

projective spaces and hyperquadrics, J. Math. Kyoto Univ., 13

(1973),31-47

[9] Mori, S. Threefolds whose canonical bundles are not

numerically effective, Ann. of Math., 116 (1982), 133-176

[10] Morrow, J. A survey of some results on complex Kahler

manifolds, Global Analysis, Univ. of Tokyo Press, 1969, 315-324

[11] Nakamura, I. Moishezon threefolds homeomorphic to p3,

to appear in J. Math. Soc. Japan

[12] Characterizations of p3 and hyperquadrics 03 in p4,

Proc. Japan Acad., 62, Ser. A, (1986), 230-233

B-1

[13] Peternell, T. On the rigidity problem for the complex

projective space, preprint

[ 14] Algebraic structures on certain 3-folds, Math. Ann.

274, (1986), 133-156

(15) Ueno, K. : Classification Theory of Algebraic Varieties and

Compact Complex Spaces, Lecture Notes in Math. ,439, Springer

Verlag, 1975

B-2

Iku NAKAMURA

Department of Mathematics

Hokkaido University

Sapporo, 060, Japan