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) = [D]
9
L(-D) = [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 .
Previous Page Next Page