SPECIAL DIVISOR S O N ALGEBRAI C CURVE S
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'
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 .
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
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 2:1ma p