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
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