SPECIAL DIVISOR S O N ALGEBRAIC CURVE S

11

In coordinat e fre e notatio n thi s is

4L:C^?{H\L)*)

where, for a general poin t pGC,

0Z,(P)

=

{hyperplan e o f sections s with s(p) = 0}.

The base locus B of the linear syste m \L\ is defined t o be the finite se t of points /?GC

where s(p) = 0 for all sections s G H°(L). I n case \L\ is base-point-free w e have

*2«V(D) = L,

while in general the relation

*i(0

p r

(l)) = I(-2O

is valid.

Since the sections s

t

are linearly independent th e image curve (j L(C) C

?r

i s nonde-

generate—i.e., it doe s not lie in a

Pr _ 1

. Consequently , there are (r + 1) points of C whose

images span ?

r,

an d it follow s tha t an y general set of q r 4- 1 points on C are in general

position on 0L(C). Thi s is equivalent t o the assertion in the proof o f (1.19), that an y r

general point s on C impose independent condition s on H°(L). A s we shall presentl y see , of

great interes t ar e the configurations o f points tha t ar e in special position (tri-chords , etc.) on

Suppose we now take L = K to be the canonical line bundle. Fo r any points p, qEC

the Riemann-Roc h theore m (1.25) gives

l(p) = h°(K(-p))-g + 2,

(1.31)

l(p + q) = h°(K(-p-q))-g+l.

Now l(p) = 2 for some point p if, and only if, C is the Riemann spher e P

1;

and l(p + q) = 2

for som e pair of points p, q exactly whe n ther e is a 2-to-l ma p

(1.32) C-Z+V

1;

i.e., whe n C is hyperelliptic. I t follows fro m (1.14) and (1.31) that i f g 1 then

h°(K(rP)) = g - \ ,

and if moreover C is nonhyperelliptic the n

h°(K(-p-q))=g-2.

The first o f these say s that th e canonical linear syste m \K\ is always base-point-free, an d the

second that i f C is nonhyperelliptic the n the canonical mappin g

is a 1-to-l embedding . I t is well known tha t fo r hyperelliptic curve s