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 .
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