10 P.A.GRIFFITH S

To apply this to (1.27) we first remar k that the bilinear mapping (1.26) is injective on

each factor separately . Consequently , if we set

Pri

= P ^

0

^ ) ) , P'

2

= ?(H\L 2)) an d ?

s

= ?(H°(L

1

® L 2))

then there is an induced holomorphic mapping

(1.29)At : P'1 x P' 2 - P *

that is a linear embedding on each factor, and (1.27) follows from (1.28).

Another way of looking at (1.26) is to consider (1.29) as the composition of the usual

Segre embedding

P * X P

2

— P *

2 X 2

followed by a linear projectio n

The cente r o f this projectio n i s the P

l 2 ! 2

correspondin g t o th e linear spac e

][,.®r,.e//(£,) g//°(L 2)

I

given by the kerne l o f the bilinear mappin g (1.26).

In particular, from thi s point o f vie w the proo f o f Clifford' s theore m center s aroun d

the multiplicatio n mappin g

(1.30) //

0

: H°(L) ® H°(KL-X) - • H°(K)

(when there is no possibilit y o f confusio n w e omit th e ® sign in the multiplicatio n o f line

bundles).

We will see that man y o f th e deepe r aspect s o f th e theor y o f specia l divisor s pertai n

to this mapping .

(c) Followin g the introductio n o f a little mor e terminology, we will discuss canonical

curves and giv e what i s perhaps the mai n geometri c pictur e o f specia l divisors .

If L — • C is any holomorphi c lin e bundle ove r an algebraic curve , we shall denot e b y

\L\ = ?(H°(LJ) th e complet e linear syste m o f divisor s (s) of holomorphic section s sGH°(L).

For an y effectiv e diviso r D w e have \D\ = \[D] |. I f no w

r = d i m | i | = A ° ( i ) - l ,

then upo n choosin g a basi s s

Q

, s

x

, ... , s

r

fo r H°(L) ther e i s a holomorphi c mappin g

0L: C —

?r

define d b y