Part 1. Th e Riemann-Roch theorem and special divisors

(a)W e begin by establishin g som e notations an d terminology , an d the n discussin g th e

Riemann-Roch theorem .

The words "compac t Rieman n surfac e o f genus g" an d "algebrai c curv e o f genu s g"

will be used interchangeably, and C will denot e suc h an object . In particular , any algebrai c

curve will be assumed t o b e smooth . B y a divisor, unless specifically mentione d t o th e con -

trary, we shall always mean a n effectiv e diviso r

here th e pt ar e not-necessarily-distinct point s o f C and d = deg(D) i s th e degre e o f D. W e

shall denot e by L(D) the vecto r spac e of meromorphi c function s f on C that satisf y

(/)

+

/

0;

equivalently, the pole s of/should b e no wors e than D. Th e basic question o f thi s mono -

graph is:

What can be said about the dimension 1(D) of L(D)1

This is most certainl y a classical proble m i n the theor y o f algebrai c curves ; with n o

machinery abou t al l that ca n be easil y sai d is that

(1.1) /(D ) d + 1.

PROOF.

An y functio n / E L(D) is uniquely determined , up t o a holomorphic functio n

on C and hence u p to a n additiv e constant , by its Lauren t development s aroun d th e point s

p{ E D. Q.E.D .

In case C = P 1 i s th e Rieman n spher e and D = px + • • • + p

d

i s any diviso r o f degre e

d i t i s clear that equalit y hold s in (1.1); consequently, one expect s the genu s of C to ente r

into an y deepe r understandin g o f th e problem , and fo r thi s we need t o tak e u p th e Riemann -

Roch theorem. W e will not giv e a complete proo f o f this result, but rathe r wil l discus s it i n

a manner tha t bring s ou t a certain topologica l characte r o f the theorem .

Our discussio n will be facilitated b y usin g some standard bu t nontrivia l result s abou t

compact Rieman n surface s (cf . § 2 o f [5]) . Thes e are concerning meromorphi c differential s

on C; any suc h meromorphic differentia l 0 has a local expressio n

(1.2) * = ( Z a vz\dz% a_ N*0,

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