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
= P ^
^ ) ) , P'
= ?(H\L 2)) an d ?
= ?(H°(L
® 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
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)
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) //
: 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
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
, s
, ... , s
fo r H°(L) ther e i s a holomorphi c mappin g
0L: C
define d b y
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