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.)