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,
3
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