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
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