SPECIAL DIVISOR S O N ALGEBRAI C CURVE S 7 In particular, fo r th e dimensio n r(D) of the complet e linea r syste m |D | the Riemann-Roc h theorem (1.14 ) gives (1.18) r(D) = d-g + i(P). As mentioned i n the previou s section, our mai n proble m i s to stud y r(D), and fo r thi s the followin g remar k i s useful : (1.19) We have r(D) r if, and only if, there is a divisor D' E \D\ that passes through any r points of C. Indeed, the conditio n tha t a divisor D' £ D pas s through a point p' G C is that (1.20) f(p') = 0 where /is a meromorphic functio n wit h (/ ) = D' -D. I t i s clear the n tha t (1.20 ) define s a hyperplane i n th e vecto r spac e L(D), and that fo r r points p[, ... , p' r the equation s /(pi)="-=/GO = o define r linear condition s o n L(D). Moreover , it i s easily seen that fo r genera l point s thes e linear condition s wil l be independent (cf . §lc ) fo r clarification) , an d thi s implies (1.19). We now com e to ou r mai n objec t o f study . DEFINITION. A divisor D is said to be special in case the index of speciality i(D) 0 i.e., in case the vector space H°(£l1(-D)) of holomorphic differentials that vanish on D is nonempty. Of course, D i s automatically specia l whenever de g D g - 1 , and fo r a general suc h D we will have (1.21) r(D) = 0 by (1.17) . O n the othe r hand , if D is special then necessaril y de g D 2g - 2 , and w e shall see below that ther e is an equivalenc e (mor e precisely , a duality) between studyin g divisor s in the two degre e ranges 0 deg D g - 1 g - 1 deg D 2g - 2 . Consequently, at leas t in principle on e may restrict attentio n t o thos e in the firs t range . In this lower degre e rang e we shall not b e interested i n the generi c diviso r tha t satisfie s (1.21) , but rathe r i n th e exceptional special divisors—i.e., thos e tha t satisf y th e pai r o f condition s (1.22) £)^0 , i(D)*0. It is with these divisor s that a majority o f geometri c question s abou t curve s are concerned . Before turnin g to th e stud y o f exceptiona l specia l divisor s w e want t o giv e the mai n general fact—Clifford's theorem—concernin g al l special divisors . Fo r thi s we recall (i) tha t a canonical diviso r is the diviso r o f zeroe s of a holomorphic differentia l co, and (ii ) tha t a n algebraic curve is hyperelliptic i n cas e there is a 2:1 ma p
Previous Page Next Page