8
P. A. GRIFFITH S
the invers e image / l (p) o f a point i s then calle d a hyperelliptic diviso r D
0
. Wit h this under -
stood, w e have
(1.23)
CLIFFORD' S THEOREM .
If D is a special divisor then
r(D)degD/2.
Moreover, if equality holds then either
(i) D is a canonical divisor, or
(ii) C is hyperelliptic and D ~ kD
0
where D0 is the hyperelliptic divisor.
PROOF. T O
establish th e inequalit y i n Clifford' s theore m w e will prov e the interestin g
auxiliary fac t that , fo r an y divisor s Dj an d D
2
,
(1.24) r(D
x
+ D 2) r{D x) + r(D 2).
To see why this should b e th e cas e we set r
x
= HD^, r
2
= r(D
2
) an d le t pv . .., p
r +r
b e
any se t o f r
x
+ r
2
point s o n C. Then , by (1.19) we may fin d a divisor D[ E \D
X
I that
passes through p
v
... , p
r
an d a divisor D2 E \D
2
\ that passe s through p
r + v
. .., p
Y
, and
therefore D\ + D
2
passe s through al l these points . B y (1.19), again, we deduce (1.24).
Now since D i s special ther e i s a divisor D' o f degree 2g~2-d suc h that D + D' = (CJ)
is a canonical divisor . Moreover , we clearly have /(/)') = /(D) . Takin g Dx = D an d D2 = D'
we obtain
KD + D') = di m P(tf
°(S21))
= g - 1,
and combinin g this with (1.24) and th e Riemann-Roc h theorem (1.18) gives the two relations
*D) + i(P)g,
r(D)-i(D) = d-g.
Adding these yield s the inequalit y i n (1.23).
We refer t o § 1 o f [1] th e analysis , from th e presen t poin t o f view , of th e situatio n
when equality hold s in Clifford' s theorem , and shal l here use the proo f o f th e inequalit y t o
introduce on e o f th e fundamenta l concept s i n th e theory , the so-calle d /i
0
-map.
For thi s we shall us e the terminolog y o f line bundles, and concernin g thes e th e basi c
facts are :
(i) T o every no t necessaril y effectiv e diviso r D-i.e., D = X
i
pi ~ 2
a
qa— ther e i s asso-
ciated a line bundle [D ] suc h tha t
[Dx +D 2] = [D J ® [D 2],
[-D] = [ D ] - \
[DJ = [D
2
] oD
x
~ D
2
.
Moreover, every lin e bundl e o n th e curv e is of th e for m [D ] for some divisor D. (Becaus e of
the third relation we shall sometimes refer to the line bundle [D ] a s the divisor class of D.)
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