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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