12 P.A.GRIFFITH S
* = /*(Oplte-i)),
and th e canonica l ma p factor s a s
n
\
/ \
^ A
P1
P * " 1
7 r °
A
where t// : P 1 P g_1 i s the standar d rationa l norma l curv e o f degre e g - 1.
In the remainde r o f thi s sectio n an d i n § 2 w e shall assume tha t C is nonhyperelliptic,
and shal l identify th e curv e wit h it s canonica l image. Explicitly , choosing a basi s cjj,... , OJ
for theholomorphi c differential s o n C, the homogeneous coordinates of a point p is written
PK(P)=
I
w
iO0 •••* 0 ? ) ] .
If Pj, ... , P j ar e point s o n C , we shall denot e b y Pj pd th e linea r spa n o f thes e point s
in the spac e P^ - 1 o f th e canonica l curve . I t i s clear what thi s mean s if th e p( ar e distinct ,
and if , e.g. , p1 = = p
k
= p, the n w e take the spa n o f pk
+ 1
, ..., p
d
togethe r wit h th e
fcth osculating spac e to th e canonica l curv e at p. Equivalently , if
D =
Pl
+ '-' + p
d
,
then th e linear spa n
is the intersection o f al l hyperplanes {a ; = 0}wher e c o €H°(K(-D)). Fro m thi s it follow s
that
dim D = g - 1 - i(D) .
On the othe r han d th e Riemann-Roc h theore m (1.18) gives
r(D) = d~g + i(D).
Eliminating the inde x o f specialit y betwee n thes e tw o equation s w e arrive a t th e
Geometric form of the Riemann-Roch theorem.
(1.33) dim. D = d - l -r(D).
In words,
On a nonhyperelliptic curve C a divisor D moves in a linear system whose dimension
is exactly proportional to the failure of the points of D to be in general position on the
canonical curve.
Using (1.33) w e may, following th e classica l paper [3 ] o f Bril l and Noether , give the
expected dimensio n coun t fo r th e varietie s o f specia l divisors . Fo r thi s we denote b y C
d
the se t o f al l effectiv e divisor s D = px + + p
d
o f degre e d. I t i s well known tha t thi s
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