SPECIAL DIVISOR S O N ALGEBRAI C CURVE S 9

(ii) I f D i s an effectiv e divisor , the n ther e i s a holomorphic sectio n s

D

G H°([D])

whose diviso r (s

D

) = D. Conversely , if L i s any line bundle an d s G H°(L) i s a holomorphic

section wit h diviso r (s) = D, the n L = [D] and 5 = Xs

D

wher e X =£ 0. I t follow s tha t

L(D)^H\[D]),

\D\ s Eff°([/)]) .

The mapping in the firs t isomorphis m i s multiplication b y s^ .

(iii) Fo r an y line bundle L an d effectiv e diviso r D, i f we set

L(D) = L® [D]

9

L(-D) = L® [D]' 1

9

then

( meromorphic section s of L wit h )

H\L(±D)) =

( poles (+ ) o r zeroe s (- ) o n D. )

In particular, if K denote s th e canonica l line bundle the n

H0(K(-Dy) = H°(a l(-D))

is the spac e o f holomorphi c differential s tha t vanis h o n D.

(iv) Wit h the notatio n h°(L) = dim H°(L) th e Riemann-Roc h theore m (1.14) become s

(1.25) h°(L) = d-g+ 1 +h°(K®L-

1).

We will now use line bundles to furthe r discus s the inequalit y (1.24) encountered i n

the proo f o f Clifford' s theorem . I f Lx an d L2 ar e any tw o lin e bundles then th e ma p

(1.26) M : # ° ( £ i )® H 0{L2)- H 0{LX ® L2),

given by multiplicatio n o f sections , is basic. Wha t th e proo f o f (1.24) shows is the inequalit y

(1.27) dim(imag e y) h°(L

x

) + h°(L

2

) - 1.

It i s instructive t o alternativel y deriv e (1.27) fro m th e followin g lemma :

(1.28) Let {JL: P

2

x P

2

—

?s

be a holomorphic mapping that is linear in each factor.

Then

dim(/i(Pri

x ?

r2))ri

+r 2.

PROOF.

Sinc e ju is holomorphic an d linear o n eac h factor ,

^(0^(1))= 0

r i

O)®0

r2

0)

P p i p 2

is an ample line bundle o n P * x P

2

. Consequently , n ca n have no positive dimensiona l

fibres. Q.E.D .