6
P. A. GRIFFITH S
of holomorphic differential s c o that vanis h on //—i.e., that satisf y (CJ ) D—then the numbe r
of independent equation s (1.13) is g - /(D) . Thi s gives, finally, th e famou s Riemann-Roch
theorem:
(1.14) l(D) = d-g+l +i(D),
derived here fro m th e existenc e theore m (1.4) plus elementary topologica l observations .
As a corollary w e may complemen t th e uppe r boun d (1.1) to rea d
(1.15) d-g+ 1 l(D)d + 1.
Moreover, since for an y holomorphic differentia ] c o
deg(co) = 2 # - 2 ,
we have that i(D) = 0 in cas e de g D 2g - 1, which whe n combine d wit h th e Riemann -
Roch theore m give s the answe r
(1.16) l(D) = d-g+ 1 if de g D 7g - \
to ou r proble m fo r divisor s of larg e degree .
However, as is well known, in geometri c question s it i s most ofte n th e divisor s o f low
degree that ar e interesting, an d a t thi s stag e al l tha t w e ca n d o easil y i s t o giv e 1(D) for
generic D, a s follows :
(1.17) For a generic divisor D of degree d we have
ifdg-l,
r "
[g-d+l if
1(D) =
ifgd2g-2.
PROOF.
Fo r a generic D w e have (cf. th e proo f o f (1.19) below)
!
g - d if dg,
0 ifdg+l.
Now use the Riemann-Roc h theore m (1.14).
(b) I n this section w e shall introduce som e furthe r algebro-geometri c terminolog y an d
then begi n ou r discussio n o f specia l divisors .
Two divisor s D and D' o f the sam e degre e d ar e sai d to b e linearly equivalent, writte n
D ~ £' , if ther e i s a meromorphic functio n f on C whose diviso r is
(f) =
D'-D.
The se t o f al l divisor s D' tha t ar e linearly equivalen t t o a fixed D is a projective spac e calle d
the complet e linea r system an d denote d b y \D\. I n fact , usin g our previou s notation i t i s
clear that th e divisor s in \D\ are in a natural one-to-on e correspondenc e wit h th e lines in
the vecto r spac e L(D) so tha t
\D\ = P(Lp)) .
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