On rational surfaces I.

M E M O I R S O F T H E C O L L E G E O F S C IE N C E , U N IV E R S IT Y O F K Y O T O , S E R I E S A Vol. XXX II, Mathematics No. 3, 1960.

On

rational

_{su rfaces}

I.

Irreducible curves

o f

arithmetic genus

0

or

1

By

Masayoshi NAGATA

(Received December 10, 1959)

W e are to prove a fundamental theorem on irreducible curves o f arithmetic genus 0 or 1 on a non-singular rational surface and to show applications of the theorem to the followings :

(1) The classification of non-singular rational surfaces which

have no exceptional curves of the first kind.

(2) The classification of rational ruled surfaces.

(3) Factorization o f Cremona plane transformations.

(4) Classification of projective surfaces of degree d which are

not in any projective space of dimension d - 1 .

Results assumed to be known : Besides elementary facts on

surfaces, we need to know ( i) irreducible exceptional curves of the first kind (see Zariski [10, Part I I ] ) and (ii) the genus formula o f curves on non-singular surfaces (see § 1, (7), (I,,)).

As for the definition of surfaces, we shall employ that in the sense of Zariski [10 ].

1. Definitions and notations.

(1) We fix an algebraically closed ground field h of an arbitrary

characteristic. Points we treat are those which are rational over

k , and curves or transformations which we treat are all defined

over k.

(2) A birational correspondence is called to be natural i f it

is the identity at every biregularly corresponding points.

(3) I f 1 is an exceptional curve of the first kind on a surface

F such that F ' is the union of F - 1 and a simple point which is

dominated by every point of 1. This transformation from F onto

F ' is called the contraction (o f F ) by 1 and is denoted by conti.

(4) The quadratic transformation with a center P is denoted

b y d ilp . The product of successive quadratic transformations dilpi

( i=1 ,••• ,n ) is denoted by p n )

W e rem ark that if conti is d efin ed on a surface F and if

P=conti

E ll

then dilp•cont, is the identity.

(5) L et P be a simple point of a surface. A simple point Q

of a birationally equivalent surface such that Q dominates P is

called an

in fin itely n ea r

point o f P :

P itself is the unique infinitely near point of order zero of P.

Q is said to be infinitely near point of order n (> 1 ) o f P i f Q is a

point of di1R{ R } with an infinitely near point R of P of order n - 1 .

L et c be a curve on F . The multiplicity o f c at such a point

Q is defined to be the multiplicity o f th e proper transform o f c

under a t Q , where the

P

i a re such that Q is an

in-finitely near p o in t o f

P

i o f order n — i and such that

P

i i s an

infinitely near point of P o f order i fo r each i. The multiplicity is denoted by m(Q ; c).

( 6 ) Assume that a non-singular projective surface F dominates birationally a non-singular projective surface F'.

(to) T h e r e are points P„ ••• ,P„ such that F=d il(,,i,...,pn )(F').

These points

P

i a re called fundamental points over F ' with respect

to

F.

(6) The geometric projection (from F onto F ') is denoted by

proj.

(a ) A birational transformation T from F ' onto some surface

T ( F') is said to be F-adm issible i f ( i ) T is natural and (ii) T (F')

is dominated by F.

Observe that the product of successive F-admissible transfor-mations is F-admissible.

(7) Let c be a curve or a linear system o f curves on a

non-singular surface F . The intersection number (c , c ) is called the

g rad e o f c and is denoted by 1(c). Th e arithmetic genus o f c is denoted by g(c).

L et u s consider, fo r th e sake o f s im p lic ity , o n ly the case

1 ) T h e general c a s e can be treated similarly by virtue o f th e notion o f virtual linear system ; see a forthcoming paper, whose title will be "On rational surfaces, II".

On rational surfaces, I 353 where c is irreducible. Assume that F and F ' are as in (6) and that proj c i s a curve o r a linear system o f curves such that

mi=m (Pi ; proj c). Then :

( ) I(c)=/(proj g(c)= g(proj c ) - 1-; m

i(mi— 1)/2.

Hence, i f F ' is a projective plane and if d is the degree of proj c, then

( 6 ) I(c )= d2- 1 4 , g ( c ) = ( d2 —3d +2-1; m+/,' m

i)/2,

and therefore

( a )

3d—. mi —./(c)-2g(c)+2.

2 . Elem entary facts on ru led surfaces.

(1) A su rface F is called a ruled surface if (i ) F is a

non-singular projective surface and (ii) there is a pencil L on F such that every member o f L i s an irreducible non-singular rational curve and such that two different members o f L have no common

p oin ts. In this case, members o f L are called generators o f F.

I f o n e F has two pencils L and L ' each satisfying the above condition, then we regard F with the set of generators L to be a different ruled surface from F with th e s e t o f generators L ' (it will be shown later that such an F is biregular to the product of two projective lines).

(2) When c is a divisor of a ruled surface F with a generator

g , the intersection number (c, g ) is called the p a rtia l degree o f c (with respect to the ruling).

( 3 ) L e t P b e a point on a ruled surface F and let g be the

generator going through P . Then the transformation contd i i pur dil„

is w ell defined o n F . Furthermore, by this transformation, we get another ruled surface whose s e t o f generators is th e proper transform o f that o f F . This transformation, with regards to the

ruling, is called th e elem entary transformation with the center P

and is denoted by elm ,. If t h e product of elmpi ( i=1 , • • • , n) is

defined, then it is denoted by elm( ,,..., p,,).

W e have immediately the followings :

(v,) Partial degrees o f curves are invariant under elementary

transformations.

( 6 ) Assume that a non-singular projective surface F ' domi-nates birationally a ruled surface F and that P is an ordinary

fundamental point on F w ith respect to F ' . Then elm p is F '-admissible and, (the set of fundamental points over F with respect to F')— P+ (the contracted point from the generator going through

P )= (the set of fundamental points over elm(F) with respect to F').

(4) T h e product o f tw o projective lines C , C ' i s a ruled

surface w ith t h e s e t o f generators either {C x P '; P ' E C ' } or

{P x C'; FEC } . Therefore we define interchanging transformation,

which is denoted by V, on C x C', to be the identity map on C x C' but we interchange the ru lin g. A ruled surface which is biregular

to C x C ' is denoted, in general, by Fo. Then we define the

inter-changing transformation V on any Fo.

For an F „ generators o f V(F0) are called base lines of the Fo.

(5) Let b be a base line of an F, and let P„ ••• ,P„ be mutually

distinct points on b. Then the ruled surface elmcp1,...,pn,(F0), o r

a ruled surface F which is biregular to this is denoted by Fn in

general. elm(pi,...,po [b ], or the curve on F which corresponds to

this curve by the biregular transformation, is called the base line

o f the ruled surface.

(6) A n y projective plane is denoted by S in general. Then w e have easily :

( i) Let P, Q be distinct points on an S and let I be the line

going through P , Q . Then Tcp, 0, = contdu(p, ( 2 )(/) -dil(p,Q ) is well

de-fined on S and Tp,Q )(S ) is an Fo, whose generators and base lines

are the proper transforms o f lines going through P (or Q ) and those through Q (or P).

Hence,

(ii) I f R is a point on an F , and if g , b are the generator

and the base line going through R on the Fo, then

contR c o n tdiim

-gr contd i i REbi•dilR is w e ll d e fin e d on th e Fo, and

contR (F0) is an S (or, biregular to an S).

(iii) I f c is a divisor on an F , with a generator g and a base

line b, then c is linearly equivalent to (g, c)b +(b, c)g.

( i v ) An Fo has no more than two structures as ruled surfaces,

and therefore.

(IT) The base line b of an F , is an exceptional curve o f the

first kind and cont,, (F1) is an S . Hence, i f P is a point of an S,

then dilp S is a n F , with the set of generators dilp [the system o f lines going through P ].

O n rational surf aces, I 355

F , as follows :

L et b b e the base line of an F , and let P be a point of the

F , which is not o n b. Then th e product dilp.contb i s a natural

transformation from the F , onto another F , . This transformation is called the interchanging transformation with the center P and is denoted by V,.

We note that if a non-singular projective surface F ' dominates the F , birationally and if P is a fundamental point with respect to F ', then V , is F'-admissible.

(7) We consider some relationship among Fn's. W e rem ark

at first the following easy fact :

L e t P , Q b e points of an Fo su ch th a t th ere is n eith er a

generator nor a base line o f th e F , which goes through both P and Q . Then elm,,,Q, F , is another F , in which the proper

trans-forms of base lines going through P or Q are base lines. A s a consequence, we have easily

PROPOSITION 1. Fn does not depend on the position of the Pi on

b in (6) above but depends only on the number n.

COROLLARY. L e t P b e a p o in t o n an Fn. T h e n e lm s (Fn) is

either Fr, o r Fr_i according to w hether P is on a base line of the Fr or not.

Now we prove

PROPOSITION 2. I f n > 0 , then the base line b o f an Fn i s the

unique irreducible curve on Fr w ith negativ e grade: the grade (b, b) o f b i s —n.

PROOF. Assume that c is an irreducible curve on the Fr= F

such that (c, c) < 0 . Let -0- be the partial degree of c. Let P„ •••,P

n

be points of the Fr at sufficiently general positions and we apply

the transformation T=e1m(p1,...,,n ). L et P p be the point of T(F)

contracted from the generator going through Pi. Then each .13 1.'

is a o--pie p o in t o f T [ c ] and is on T [ b ] . Set 02 = (T [c ], T [b ]).

Since T (F ) is an F , with a base line T [b ], w e have ( T [c ], T [c ])

—20-0-'. Hence (c, c)=20-01

— no-' < 0 , which shows that 20-' <no-.

But T [ c ] and T [ b ] have common n points P ,. which are 0—pie on

T [ c ] . Hence T [ c ] and T [ b ] must coincide with each other. That (b, b)— —n is obvious. Thus Proposition 2 is proved.

COROLLARY. I f n-i=n', then an Fn c an n o t b e biregular to any

A s another consequence of Proposition 2 , we have

PROPOSITION 3. I f n + 1 , then F „ h as no exceptional curve of

th e f irs t k in d ; o n th e other hand, the base lin e o f an F , i s the unique exceptional curve of the f irst k ind on the Fi.

3 . A numerical lemma.

PROPOSITION 4. A ssum e t h a t d , m „ ••• , mr a r e non-negative integers such that ( i) m i>m 2 >• • • >m r, (ii) ( 2 i < t < r ) ,

( ii i ) m1+m2< d , an d (iv) setting s = 3 d - m „ d 2-1 ' InK sm t+i•

T hen w e have m1+2m1+1> d , except f o r th e follow ing three cases:

( 1 ) d =m „ m2= 0 ; (2 ) d =3 m „ t =2 , m- 1 - -m u, mu+i- 0 , 3 < u ;

(3) t = 2 , d =2 m1, m 1=M 2 m3= • • • = M u = d 14, ma + 1=0 , 3 < u .

PROOF. Set w = m , - / m i. Then (v) w > 0 , ( v i )

(v ii) / 4 , mi = 3 d - s - 2 m ,+ w . Now we consider the condition (iv) :

d2_{- s m} t, < T W < m 7 + m 2(:V2m1) +m1-_F1( ç+1m1) =m 7+m 1m 2-w m , +(3d - 2 mi)mt+i - _{smt+i + wm} 1 +1=m1(m, +m,)+ (3d - _2mi)mi+i - w(m2- _m1+1)- smt+i• B y (v ), w e have

(viii) cl2 _{<m i(m i+ m 0+ (3d} - _{2mi)mt Fi,}

here, the equality implies / t2m .=m2( mi) and Irt+, m7=mt-1-1(I4imi),

namely, i f mi + 0 , then mi is e q u a l to m , o r m1,1 according to

whether 2 < i < t o r i > t + 1 .

(I) I f 2m1> d , th en ( v i i i ) shows th a t d2< m 1d+2dm1 + 1

-d(m1+2mt+1) and that the equality holds only if m = 0 .

I f In, „ = 0 , then w e have mi= d , hence by (iii), it is the case (1).

I f mt„ I 0 , then th e above is strictly inequality and w e have

m1+2

m1+1>d.

(II) Assume now that 2m1< d and set cr = (d/2)- m1.

Then 0-> 0 . ( v i i i ) shows that d2< m

1(d -2 0-+ m 2 ) + ( 2 d + 2 o

-)m1+1

=d(m1+2mt+1)- 2a-_(ml- _m1+1)- m1(mi -m2). Hence we have m1+2_m1+1

> d ; I f m 1+2m1 1= d , then w e have 0

-(m,

-m t+ ,)= 0, -ml=m , and

the facts remarked after (viii). If 0-= 0, then we have the case (3 ); i f ad = 0 then we have the case (2). Thus the proof is completed.

COROLLARY. A ssume that d, m1, ••• are non-negative integers

w hich satisf y ( i) , ( i i ) an d ( i i i ) above. S e t g =( d2_{- 3 d + 2 - / m7+}

m1)/ 2 an d I = d 2- 1 m . I f g < 1 a n d i f I + 2 - 2 g > 0 , then we

hav e m 1 + 2 m t i - 1 > d , e x c e p t f o r th e f ollow ing s ix c a s e s : ( 1 ) g <0 ,

On rational surf aces, I 357

d = 3 , non-zero m , are 1 and the number of such mi's is at m o st 9;

(5) g =1 , d =3 m „ t =2, m-1 • , 174, 1 — 0, m 1

d= 1, u = 9 ; (6) g =1 ,

d =2 m1, m 1 =m 3 , m 3 = ••• =m u =d I4 , T 1 2 ,1 = 0 , 2 < u < 10 ; h e re , if

d + 4 , then u =1 0 .

P R O O F . Since 3d— mi = /— 2g+ 2, we see that d2 -

ti4 <

3d

(tf) If mi , , > 1 , then the conditions in Proposition 4 are

satis-fied, hence we have in1+ 2 m , , , > d except fo r th e three cases (1),

(2), (3) in Proposition 4.

By our assumption on g and /, w e have our assertion easily

in this case.

( 6 ) Assume now that m ,1= 0 . I f I = d21 ) 2 < 0, then

Pro-position 4 is applied and w e have ou r assertion. Assume now

t h a t 1 > 0 . (i) Assume that m2=1-0. T h e n w e m a y assume that

mtd= 0- T h e n the integers d, m i, ••• , m t, 1 satisfies the conditions

in Proposition 4 w ith th e same t and therefore we have either

m, + 2 > d or d = 3, t =2, m , = m2= 1 o r d = 4 , t = 2 , m, =m2= 2 ;

whence we have our assertion in this case. (ii) Assume that m2=0.

W e want to prove that either d < 3 o r m ,>d — 1. Assume the

contrary. T h e n 0 - = d— m, i s n o t l e s s t h a n 2 . W e have

> 2g — 2 =d2 — / / 4 + m i = 2 0 - m , (72— 2m, 30- = 2m, (0- 1) F cr(a.— 3).

T h e r e fo r e a < 3 a n d th e equality implies m, = 0, hence d = 3.

Therefore 0-2 by our assumption and w e have 2m1 — 2<0, i.e.,

m ,< 1 a n d d < 3 . Thus the proof is completed.

4 . A fundamental theorem.

W e are to prove in this section the following theorem which is fundamental throughout this paper :

THEOREM 1. A ssum e th at a non-singular surf ace F dominates

b iratio n ally a n F„,2) an d le t c be a n irreducible curv e" on F such

2) A s a consequence of our Theorem 2 in § 5 b elo w , w e see th at a non-singular ra tio n a l surface F dom inates birationally an F „ if an d o n ly if F is n o t biregular to a n y S.

3) It is obvious that our T heorem 1 can be applied to an irreducible linear system of curves on F sa tisfy in g the conditions on the genus and the grade.

th at eith er

g ( c ) = 0

an d

I(c) > - 2

o r

g (c)= 1

an d

I(c) > 0 . "

Then there is a n

F -adm issible

transf orm ation

T ,

w hich is the product of suitable,

F -adm issible

elem entary o r interchanging transformations, su c h th at the projection

c '

o f

c

o n _{T ( F , z ) ,} w h ic h is an Fr, satisfies one of the follow ing conditions; here, a. is the partial degree o f

c'

and

b

is a base line o f t h e r

(I) W hen

g(c)= 0 :

c '

is a point.

( 6 ) c '

is a generator and

c '

does not g o through any f

unda-m ental point w ith respect to F.

( a )

o-= 1 ,

c '

does not go through any fundam ental points w ith respect to

F

an d

I(c)= 2(c',b)+ r.

(vc) r = 1 ,

o-= 2 ,

(c', b)= 0, I(c)= 4

an d

c '

does not g o through

any f undam ental point w ith respect to F. (II) W hen

g(c)= 1 :

(

a )

r= 1

an d

cont

b

[ c l

is a cubic curve which goes through at

m ost

9

f undam ental p o in ts w ith respect to F; these f undam ental points are sim ple on

cont

b

[ c l .

(--\) r = 1

an d

cont

b

[ c l

is a curv e o f degree

3 f (t> 2 )

which

goes through ex actly

9

f undam ental points w ith respect to

F ;

these f undam ental points are ex actly

t-

pie o n

cont

b

[ c l .

(L ) I (c)= 8 , c'

does not go through any fundam ental point with

respect to F , Cr = 2 an d

r

is equal to either 0 o r 2.

In

order to prove our theorem , it is sufficient to prove

the

following

:

L E M M A .

With

the notations in

Theorem

1 , le t P „ • • , P , ,

be

fundamental

points

over F

n

w ith

respect

to F ,

and let mi

b e

the

multiplicity

o f

c ' a t

Pi ;

w e m ay

assume

that

mi> m ,> ••• >mt.

(i)

B y

successive

elementary

transformations

with centers

Pi

such that

mi>0-/2,

w e can reduce to

the case

w h ere all m

i are

at m o st 012 (n

in the

last

situation

may be different from

the

n

a t

the begining).

(ii) A t t h e

reduced

situation in ( i )

_{w e}

have one of the

following cases

:

( 4 ) O n e c a s e o f

those

in

Theorem

1.

4 ) A special c a se o f our Theorem 1 for the case where g ( c ) = 0 and / (c )= —1 (i.e ., c is an exceptional curve of the first kind) was proved independently by M r. Knap ; he proved independently our Theorem 2 by virtue of the special case of Theo-rem 1.

On ratio n al surfaces, I 359

(12) n = 1 and, for a suitable P „ the partial degree o f V , [ c l

is less than o-.

(,-) n = 0 and the partial degree o f V [ c '] is less than cr.

) n = 0 o r 2 a n d , fo r a suitable P„ e lm pi [ c l is in the

case ( n ) or (-\) in Theorem 1.

PROOF. ( i ) is obviou s and w e prove (ii) : W e assume that

cr> 1 .

( y ) When n > 2 : L e t Q„ ••• , Q„_, be points of F „ at

suffi-ciently general positions. Then elm( _ i ) (Fa) is an F , . Let

be the point on the F, contracted from the generator going through

Q1. T h e n (21` are o--ple points of the proper transform c * o f c'

on the F , . L et b* be the base line of the F , and set 0-*=(c*, b*).

B y the transformation contb* from the F , onto an S , c* is

trans-formed to a curve c " o f degree c + o -* and the contracted point P*

from b * is a 0-*-ple p oin t of c". Since Q t are on b*, u* is not

less than ( n - 1 ) Π. N ow w e apply th e corollary to Proposition 4

(for d =u +Œ * ,Œ * ,Œ , « •, m „ ••• , mi.) ; since mi<a-I2, o-* +2m ,<d,

and w e have one of the following cases :

( i ) g = 1 , Œ =Œ * , n -2 , either o- =2, = 0 o r 0-= 2 m 1, 1 = • • • m s ,m ms, =0 (1 < s < 7 ; i f m1> 1 , then s =7 ) , (ii) g =0 , o- --= 1, hence mi =O. ( i ) gives either ( L ) or ) according to whether m , is equal to zero

or not ; ( i i ) gives (a ) because /(c)=(0-+0-*)2-0-*2—(n—l)cr2 = 2o-* —

n +2 =2 [(c', b)+ n —1 ] —n + 2 =2(c', b)+n.

( L ) When n - 0 : We can do the same as above for n - 2 and

the difference is that Q t is not on the base line b * . The partial

degree o f V [c '] is equ al to 0-*. Assume that cr*> a-. Then we

apply the corollary to Proposition 4 fo r d=o-+Œ*, m „ • • • , ;Tit

and w e have the assertion similarly.

(N ) When n = 1 : L e t b be the base line of the F , and

con-sider contb. Set 0-* =(c', b). Then c o n t, [c l is a curve o f degree

d— Œ+Œ* on the projective plane S=cont, (F1).

(i) I f mi> e , then w e m ay assume that P , is an ordinary

point of the S and w e see th at the partial degree of V ,1 [c '] is

equal to cr+o-* — m „ which is less than Œ; thus we have the case

(0).

(ii) Assume now that

e>m ,.

Then we apply the corollary

to Proposition 4 fo r d =0-+Œ*,Œ*,m „ ••• , m, and w e have one of

the following cases :

is contained

in

_(a).

(ii) g=0, d =

2,

o-*=0 ;

this is

the case (rc).

(iii) The case

(a).

(iv) The case

(--\).

Thus

the

proof is completed.

5 . _{The classification of rational surfaces which have no}

excep-tional cu rve o f th e first kind.

THEOREM

2.

5 )

."

A non-singular rational projective surface F

has no exceptional curve of the f irst k ind if and only i f F is either

an F„ w ith n + 1 or biregular to an S.

PROOF.

I f p a r t

w as p roved already

in Proposition

3.

We

prove

the

only

if part. Assume the

_{contrary. Then there exists}

a

non-singular rational

projective surface F

which does not

do-minate

birationally

any

o f S and F .

Since there is

a dilatation

T o f F

such that

T (F )

dominates some

S or Fr,

w e m ay

assume

th a t th e re is

a point P on F

such that

F*

= dil,

(F )

dominates

some

S or F . L e t c*

be

dilp

P . Assume

_{th at}

F *

dominates

an

S. F *

cannot be

the S

because

c*

is

an

_{exceptional curve}

of the

first k in d . T h e re fo re th e re is

a

_point

Q o n th e S

_{which is}

fundamental with

respect

to

F * ,

i.e.,

F *

dominates

dil

o (S),

which

is

an F , .

Thus

F *

dominates some F.

in

any

ca se. L et c '

be

the

transform

o f c * o n

th e

F „ .

B y T heorem

1 ,

w e m a y

assume

that

c'

satisfies

one of the conditions in

Theorem

1 ,

(I). I f c '

is

a point,

then

F

dominates

the F„,

which is

a contradiction,

_hence

it cannot be

the case

0,

,

).

Since

/(c*)=

—1,

it cannot be

the case

( 6 )

nor

(vL).

T hus it

must

be

th e ca se (a ).

Since

/(c*)=

—1<0,

the intersection

number (c', b)

o f c '

with

the base

_line

b of the F„

must

be negative, i.e.,

r + 0 and c'= b.

Therefore —1

=1 =

—2n

+n

= —n

which

shows

that

n = 1,

hence

c'

is

an

exceptional curve

of the

first kind

and F

dominates

cont,

,

F , w h ic h is

an S : This

is

a contradiction.

Thus

the

proof is completed.

5) See note 4).

6) A very difficult proof o f our Theorem 2 for the classical case was given by Andreotti [2]. (T h e r e is an error in A n d reo tti's last statement ; he proved correctly that such a surface is biregular to either a projective plane or a ruled surface, and he mis-stated the converse and he excluded F2„ +1 fo r a ll n=0, 1, 2, •••.)

On rational surfaces, I 361

6. T h e Classification o f rational ru led surfaces.

PROPOSITION 5. I f F is a rational ruled surface, th e n F is an

F„ w ith an n. If, f urtherm ore, F has another structure as a ruled

surface, th e n n =0 and h as no more such structure.

PROOF. L et F ' be an Fn which is birationally dominated by F.

Let L be the set of generators o f F and let L ' b e th e transform

o f L on F ' . L e t OE be the partial degree o f L '. Since g(L )=0,

(L, L) = 0 , w e m a y assume, by virtue o f Theorem 1, that either

L ' is the set of generators o f F ' or the partial degree 0- o f L ' is

one and L ' does not go through any fundamental point with respect

t o F and 0=(L , L )= 2 (L ', b ) +n , here b is the base line o f F'.

Hence n must be zero and (L ', b )=0 , which shows that L ' is the

s e t o f generators o f V ( F ') . Thus we may assume that L ' is the

s e t o f generators o f F ', and w e see th at F '= F because every

member o f L is irreducible. Assume now that a pencil L * on F,

different from L , defines a structure of ruled surface on F . Since

F=F', L * must satisfy the assumption stated above. Since L--+L*,

L * must b e th e s e t o f generators o f V (F'), n being z e r o . Thus

the proof is completed.

7. A rem ark o n s im p le p o in ts o f ration al surfaces.

As an application of Theorem 2, we prove the following :

THEOREM 3. L e t F be a giv en projective ratio n al s u rfa ce . If

, P„' are giv en simple points on a rational projectiv e surface

F ', then there is a birational m apping T f rom F' onto F such that T is b ire g u lar at e ac h P.

PROOF. We may assume that F and F ' are non-singular.

Then F and F ' dominates birationally G and G ' respectively, each

o f which is either a projective plane or an _{F .}

I f G ' is not an S , then taking suitable points of F 'A G ', say

Q ; , • • • , , w e can m ake or,, G ', which is dominated

by ( 2 / , F ', t o b e a n F , . T h e n w e m a y replace F ' by

dil( Q i,,.,., Q/) (F'), because the .FTs are on this last surface. Thus

w e m a y assume th a t G ' i s a n S. I f G is an S , then, taking

fundamental points Q , o f G w ith respect t o F , we consider a

birational transformation from G' onto G such that no P is mapped

to any _{Q .} and w e have the required transformation easily : I f G

elementary transformations on the Fr a s above for F ' and G', we

can reduce to the case where G is an S ; as for the transformation

from G ' onto the new S, we only require that no I3 is mapped to

any _{Q . o r to any point of the curves came from some points of}

G . Thus the proof is completed.

8. Some sim ple applications of Theorem 1.

A s another application of Theorem 1 , we have the following amusing result immediately by virtue o f Theorem 2.

THEOREM 4. I f c is an irreducible curve o f arithm etic genus 1

on a non-singular rational surface, then the grade of c is at m ost 9. On the other hand, w e have

THEOREM 5. L e t F b e a non-singular rational surface which

dom inates birationally an Fn and assume that there are at m ost h

ordinary f undam ental p o in ts o f Fn w i t h respect to F . I f F has

more than 2h irreducible ex ceptional curv es of the first k ind, then F

dominates an S.

PROOF. T h e assumption shows that there is an exceptional

curve c of the first kind on F whose projection on F„ is a curve.

Then we apply Theorem 1 to this c and we see that F dominates

an F „ hence an S, too.

9. Factorization of Crem ona plane transformations.

(1) A Cremona transformation is a birational transformation

from a projective space to a projective space and w e are to deal only with the case of projective planes.

It is obvious that a Cremona transformation is the product of a natural Cremona transformation and a linear transformation and therefore we deal with natural Cremona transformations in this section.

(2) L e t T be a Cremona transformation from an S to another

S , s a y S'. L e t F b e a non-singular surface which dominates

birationally both S and S'. Let L ' be th e linear system o f lines on S' and let L*, L be the transforms of L on F and S respectively. It is obvious that g(L*)=0, A L*)=1.

We denote in this section L and L * b y i(T ) and 1*(T)

respec-tiv e ly ; th o u g h *(T) depends really on th e choice o f F , we do

O n rational surf aces, I 363

(3) A Cremona transformation T is called a Jon quières

trans-f orm ation if i( T ) has a base point P such that the multiplicity of

i ( T ) at P is one less than the degree o f th e members o f I( T) ;

i f I( T ) consists o f conics, then T is called a quadratic Cremona

transformation.

It is obvious that T is a quadratic Cremona transformation if

and only if it is a Cremona transformation with three fundamental

points ( =base points of I( T)).

(4) Now we can state the factorization theorem o f Cremona

transformations as follows :

THEOREM 6 (Noether-Castelnuovo's theorem).7 ) W ith the notations

i n (2), th e Crem ona transf orm ation T i s th e p ro d u ct o f a finite number of F-adm issible Jon quières transform ations ; each Jon quières transf orm ation is the product of successive quadratic Cremona trans-formations.

(5) In order to prove Theorem 6, making use of Theorem 1,

we interprete T into a transformation from an F, onto another F,

as follows :

Let P and P ' be fundamental points of S and S ' with respect

t o S ' and S respectively. Then T i s th e product of dilp, the

natural transformation from dilp (S ) onto dilp, (S ') and the inverse

of dilp,.

Let L " be the transform o f L on dilp (S).

B y the definition of Jonquières transformations, w e have

L E M M A . If the partial degree of L" is 1, then T is a Jonquières

transformation. Conversely, i f T is a Jonquières transformation

a n d if P is s u c h th a t (T ) has maximal multiplicity at P , then

the partial degree o f L " is 1.

A s a consequence, we have

C O R O L L A R Y . If the transformation from dilp (S ) onto dilp, (5)

is the product of successive elementary transformations, then T is

a Jonquières transformation.

7 ) The usual statement of th e factorization theorem is that T is the product of quadratic Cremona transformations, which was claimed at first by Noether [7] (c f. [8 ]) but unfortunately the proof contained errors. The factorization was proved by Castel-nuovo [4] at the first time ; he proved our statement (not explicitly). Alexander [1] objected to Castelnuovo's proof by his misunderstandings o f infinitely near points and gave a new proof of the factorization into the quadratic ones. Later in 1939, Jung [5] gave a nicer proof using Fo.

( 6 ) The above corollary and the following easy facts shows the validity of the first half o f Theorem 6:

(to) B y Theorem 1, applied IN T ) and dil, (S), shows that the transformation from dilp(S) to dilp,(S') is the product of elementary transformations and interchanging transformations, hence it can be factored to the product o f following types of transformations.

(i) From an F, to another F, only by F-admissible elementary

transformations.

(ii) From an F1 to another F , only by an F-admissible

inter-changing transformation.

(iii) From an F , to another F , through an F0 ; (elementary

transformation). V. (elementary transformation).

(6) In the above transformations, that of type ( i) corresponds

to a Jonquières transformation by the corollary to the lemma in (5) ; that of type (ii) corresponds to the identity Cremona transfor-mation, as is obvious ; that of type (iii) corresponds obviously to a quadratic Cremona transformation, by virtue of the remark given at the end of (3).

(7) Now we prove the last half of Theorem 6. B y the lemma

in (5) (converse part), w e assume that L " is of partial degree 1 and we are to prove that T is the product o f a finite number of quadratic Cremona transformations.

We prove the assertion by double induction on the number t

o f fundamental points ( = base points of L " ) and the minimum m o f orders o f infinitely near points which a r e fundamental and which does not lie on the base line.

(i) I f m =0, i.e., i f there is an ordinary fundamental point

Q which is not on the base line o f dilp (S ), then we apply elmQ

and w e have an Fo and therefore taking a fundamental point Q'

such that elm( Q,Q,) is well defined, elm( Q,Q,) (dil, (S )) is an F , on

which there a r e two less fundamental points w ith respect to dilp, (S'), and this case is proved.

(ii) Assume now that m > 0 and let Q be a point at sufficiently

general position and let Q' be an ordinary fundamental point such

that there is an infinitely near point Q * o f Q ' o f order m which

does not lie on the base line and such that Q* is fundamental.

Then elm( Q ,Q,) (dil, (S)) is an F „ which has the same number

o f fundamental points w ith respect to dilp, (S ') and on which Q* is an infinitely near poin t of order m - 1 . H en ce, by induction assumption, we prove this case.

O n rational surf aces, 1 365

1 0 . P ro jective su rfa ces o f d eg ree d which a r e n o t in an y S a.

The following lemma is easy and is well known :

LEMMA 1. I f general members of hyper-plane sections of a surface F are rational curves, then F is a rational surface.

Now we prove

LEMMA 2. If a projective surface F o f degree d is not in any

projective space Sd of dimension d , then F is rational.

Furthermore let c be an irreducible member o f th e system L

of hyper-plane sections of F . Then c is a non-singular rational

curve and L is complete.

PROOF. The trace Lc o f L o n c is o f degree d and of

dimen-sion d , which implies our assertions.

THEOREM 7." W ith the notations as above, F is either a cone"

w ith non-singular rational base curv e o f degree d i n Sd _{o r a n F.}

s u c h th at d - 3 - - n is a non-negative even number an d such that the generators are lines, o r d = 4 an d F is biregular to an S ; F is a

V eronese transform of an S .

PROOF. Let F ' be a non-singular rational surface which

domi-nates F birationally and such that F ' is not biregular to any S.

Let L ' be the transform of the hyper-plane sections L of F on F'.

Since F ' dominates some F „ , we can apply Theorem 1, because

g (L )= 0 by Lemma 2 and we may assume that the projection L*

of L' on F„ is in one case in the conclusion of Theorem 1. Since L * defines a birational transformation, L* cannot be the system

of generators, and we see that the partial degree 0- o f L * is either

1 or 2 an d that (ts) if 0 --1 , then d =2(L* , b)+ n and L * has no

base point ; (6) if tr= 2, then d=4, n=1, (L*, b)=0.

In the case L * is substancially the system o f conics on

an S, hence F is biregular to S and d =4.

Now we consider the case (i,). Since L is complete, we see

th at I , * is com plete an d therefore I,* = Ib + mg1 where g is a

generator of the F „ . Since (b, L*) >0, we have m > n . Since o- =l,

any member o f L * going through two points on a generator g '

contains g', h en ce L* — g' has dimension two less than dim L*,

which shows that the corresponding curve on F to g ' must be a

line ; namely, the images of generators are lines.

8) Th is was proved by Del Pezzo (see B ertini [3; Chap. 15, § 9]). 9) We include the case where d =1 , and F is a projective plane.

(i) When (L*, b ) =0 , i.e., m =n — d :

The base line corresponds to a point on F. Since the generators

are mapped to lines, we see that F must be a con e. A base curve is an irreducible hyper-plane section, hence it is a non-singular rational curve by Lemma 2.

W e rem ark by the way that since there are such cones for all d =1 , 2, ••• , we see that

PROPOSITION 6. T h e com plete linear sy stem jb +m g l o n Fn i s

irreducible f o r an y m > n , w here b an d g are a b as e lin e an d a generator respectively.

(ii) Now we consider the case (L*, b) > O , i.e., m > n :

Since ib +n g i gives the cone as above, whose quadratic dilatation

with vertex as the center gives the F „, we see obviously that

PROPOSITION 7. If m >n , then the complete linear system Ib+m gj

defines a b ire g u lar im bedding o f th e F „ i n S d+' as a s u rf ac e o f degree d , w here d=n+1+2 (m — n ).

This Proposition 6 proves that F is an F . Thus the proof

o f Theorem 7 is completed.

REMARK. It is easy, by virtue of Proposition 7, to see the

following : W hen d =n+ 3, i.e., F is defined by I b +(n +1 )g l, the

base line is imbedded as a line.

1 1 . Su rfaces of d e g r e e d in S a bu t n o t in a n y S .

W e are to prove at first the following

THEOREM 8. If a p ro je c tiv e s u rf ac e F o f degree d i s in a

projective space S d of dim ension d but not in an y hy per-plane Sd- i,

then F is one of the follow ings:

(1,,) A p ro jectio n o f o n e in Theorem 7 w ith center outside of

the surface.

( 6 ) T he system L of hy per-plane sections of F is represented on an S as a sy stem of cubic curv es w ith at m ost 6 base points and whose general members are non-singular cubic curves.

(a ) d = 8 an d F is an Fo; V eronese transform o f F , in S3.

(vc) d = 8 an d F is biregular to a cone in S ' with non-singular plane conic as a b ase curv e; V eronese transform o f th e cone.

(a ) F is a cone w ith a non-singular elliptic base curve.

The first step o f our proof is to prove that

On rational surfaces, I 367

This assertion is obvious i f F has a singular point P ,

con-sidering the subsystem of L of dimension 2 having P and other d -3

base points. Therefore w e assume that F is non-singular. We

prove the assertion b y induction on d.

I f d = 3 , then F carries a line, say 1; for the proof, see, for

instance Waerden [9] (w e do not need to prove th e finiteness of

the number o f lines). Considering plane sections of F containing

1, w e see that F carries a linear pencil o f conics, which implies

that F is a rational surface.

Now we assume that the assertion is true for those of smaller

degrees. L e t P b e a p oin t on F a n d let L * b e th e system of

hyper-plane sections going through P . Then dim L *— d -1 and

the surface F * defined by L * has degree d ' which is a factor of

d - 1 . Since dim L * = d -1 , F * cannot be in Sd - 2, and therefore

d '= d -1 (cf. th e proof o f Lemma 2 in § 10), which implies that

(i) F * is birational with F and (ii) F * is either a rational surface

o f a n irrational cone. Let us assume th at F * is n o t rational,

hence is an irrational cone. Set F **= d il, (F ) and let L** be the

transform o f L * o n F * * ; L * * has no base points and F * is

dominated by F**.

Now we need the following lemma :

PROPOSITION 8 . Assume that a non-singular surface F ' dominates

an irrational cone with the vertex P , then there is a n irrational component of the total transforms of P in F'.

PROOF. We may assume that F dominates the quadratic

dilata-tion of the cone with the center P, because exceptional curves of

the first kinds are rational. Since d il, {P } is biregular to a base curve, we have the assertion.

Now, let E be an irrational component of the total transform

o f th e v e r te x Q o f t h e cone F * in F * * a n d le t L " b e the

subsystem o f L * * which contains E . Then the members o f L"

corresponds to hyper-plane sections o f F * going through Q, each

o f which has d - 1 components. Therefore each of members o f L"

has at least d components, one of them being E . Therefore general

members of the projection of L" on F has at least d components.

Since F is of degree d, we see that every such components are lines,

this is impossible because E is irrational. Therefore F is rational

and (* ) is proved completely.

Before coming to the second step, we need to know the special case where g '= 1 of the following lemma :

PROPOSITION 9. I f L i s a com plete linear sy stem o f degree d and of dimension cl— g' on a non-singular curv e C o f genus g and i f d > 2 g '+1 , then L is non-special and g =g '.

PROOF. I f L is non-special, then we have g = g '. Therefore

we assume that L is special ; let K b e th e canonical system and

let D be a member o f L . Then

dim L =d — e =d — g + 1+ dim 1 K— D ,

hence dim 1K—Dl = g— g' —1.

Since L is special, we have dim K>dim L + dim 1K—DI,

i.e., g-1> d — g' + g— g' — 1, i.e., d < 2g', which is a contradiction,

and the proposition is proved.

Now, applying Proposition 8 to the trace of L on an irreducible

member of L , we see that these members are either rational curves

o r elliptic curves ; in th e last case, they must be non-singular

because d > 3 . Therefore we have that if F is an irrational cone,

then a base curve is a non-singular elliptic curve. Therefore we treat on ly w ith ration al case fro m n o w o n . W e im itate the

notations L , F', F„, L ', L *.

I f L is not complete, then we see obviously that F is of type

(i) and therefore we assume that L is complete. Since g(L )<1,

we can apply Theorem 1 and we may assume that L* satisfies the

conclusion in Theorem 1. If g(L )=O, then we see that dim L = d+ 1

by virtue o f Theorem 1. Therefore g(L ) must be 1. Hence L is

one of the following types : ( i ) n =1 and L can be mapped to a

system of cubic curves with at most 6 base points (because d > 3 );

(ii) n=0, cr =2, (L*, b) =2, I=d =8 ; (iii) n=2, d=8, =2, (L*, b)=0.

( i ) gives the type (6), (ii) gives (a ) and (iii) gives (c). Thus Theorem 8 is proved completely.

N o w w e observe some properties o f those surfaces stated in Theorem 8.

It is obvious that those in Theorem 8 , ( ) m ay o r may not be non-singular (they are not arithmetically normal) and that they carry infinitely many lines, except fo r those which are projections of surfaces of degree 4 which are Veronese transforms o f some projective plane S.

I f d= 3, then the surfaces of this type have singular loci which

are lines ; this fact follows immediately from the following results :

PROPOSITION 1 0 . If a s u r fa c e F a s in T heorem 8 has tw o

distinct sin g u lar Points P and Q , then the line I going through P

On rational surf aces, I 369

PROOF. Assume the contrary and let R„ ••• be points on

F at sufficiently general positions and let H b e th e system of

hyper-plane of Sd going through P, Q, R„ ••• Two general

members H , and H , c u t out hyper-plane sections of F whose

intersection contains 2P+ 2Q + Ri, which is a contradiction because

F is o f degree d.

COROLLARY. If a surface F as in Theorem 8 is not normal,

then the singular locus of F is a line.

REMARK 1. Such a n F as above corollary is o f ty p e (o ) in

Theorem 8, because those o f other types are normal.

REMARK 2. I f d = 3 , then such a n F a s in Theorem 8 are

hyper-surface, hence th e normality and the arithmetic normality are equivalent to each other ; therefore those of typ e (o) a re not normal in this case.

We consider those o f typ e (6 ) in Theorem 8: It is easy to see that")

PROPOSITION 11. L e t P„ ••• ( 0 < s < 6 ) be points such that

is w ell def ined o n a n S . T hen the system L * o f cubic curves on the S w ith pre-assigned base Points P „ , P , represents such an L as in Theorem 8, ( a ) , if and only if th e Pi satisf ies the follow ing tw o conditions:

(i) A ny four points am ong the Pi a re not colinear.

(ii) F or each j, d ilpi Pi carries at m o st one of the Pi.

Now we prove

THEOREM 9. Assume that L * above in Proposition 11 represents

such an L o f F in Theorem 8 , ( 6 ) . Then

( 4 ) A n irreducible div isor 1 on F is a line if and only if it

is the proper image of either a line on the S going through exactly tw o of the Pi o r one o f the. Pi o r a conic going through ex actly f iv e of the Pi.

(T:1) F is non-singular if a n d only i f ( i ) the Pi a re ordinary

points of the S , (ii) any three points am ong the Pi a r e not colinear

and (iii) there is no conic carry ing six of the Pi.

PROOF. Set

( 4 ) W hen 1 is th e proper im age of a curve on the S , then

we see easily that the assertion is true in this case, only because

10) A foundation of th e theory o f linear systems with pre-assigned base

1 is a line if and only if dim L -1 1 = d im L -2 . Assume that 1 is

a component of the image of a point on the S ; such a point must

be one of the Pi , say P . L et P„ be the ordinary point of S of

which Pi is an infinitely near p o in t. Then the subsystem L * * of

L * having P„ as a double point has dimension at least dim L * 2 ,

hence th e proper im a ge o f Pu is e ith e r a line or a point. The

transform pu o f Pu in F * contains that p , o f P „ If the transform

o f P„ is a point, then every Pt is not fundamental with respect to

F, which is not the case. Hence, applying it to t = u as a particular

case, the im age of P „ in F is a line, and therefore P „ and Pt

have the same image in F, which is a lin e . Thus (4) is proved.

(T3) I f p a r t is obvious, because, in that case, F is biregular

to F * . I f some Pi is a n infinitely near point of another Pi, say

P1, then the proper image of P, in F * is reducible and its image in

F is a lin e . Therefore we see that, since the Pi are fundamental

with respect t o F , the total transform of a point on F which is

the proper image of dilp, P „ is not an exceptional divisor of the

first kind ; hence F carries a singular p o in t. It is obvious that if

there are three colinear points among the Pi o r if there are six Pi

which are on a conic, then F has a singular p oin t. Thus (12) is

proved.

REMARK. A particular case where s =6, i.e., d=3, of Theorem 9

shows what are the 27 lines on a non-singular cubic surfaces in S'.

REFERENCES

[ 1 ] J. W. Alexander, On the factorization o f Cremona plane transformations, Trans. Amer. Math. Soc., 17 (1916), pp. 295-300.

[ 2 ] A . A ndreotti, O n the complex structures of a class of simply-connectedr mani-folds, Algebraic Geometry and Topology, Princeton Univ. Press (1957), pp. 53-77. [ 3 ] E. Bertini, Einfiihrung in die projective Geometrie mehrdimensionaler Raume,

Wien (1924) ; or its Italian edition.

[ 4 ] G. Castelnuovo, La trasformazioni generatrici del gruppo Cremoniano nel piano, Atti Accad. Sci. Torino, 36 (1901), pp. 861-874.

[ 5 ] H. W . E. Jung, Zusammensetzung von Cremonatransformationen der Ebene ans quadratischen Transformationen, Crelle's J., 180 (1939), pp. 97-109.

[ 6 ] H. P. Hudson, Cremona transformations, Cambridge Univ. Press (1927). [ 7 ] M . N oether, Ober die auf Ebenen eindeutig abbildbaren algebraischen Flâchen,

Gottingen Nachr. (1870), pp. 1-6.

[ 8 ] M. Noether, Zur Theorie der eindeutigen Ebenentransformationen, Math. Ann., 5 (1872), pp. 635-639.

[ 9 ] B. L . van der Waerden, Einfohrung in die algebraische Geometrie, Springer, Berlin (1939).

[ 1 0 ] O . Z ariski, Introduction t o th e problem of m in im al models in th e theory of algebraic surfaces, Publ. Math. Soc. Japan, 4 (1958).