Preface
During the perio d Ma y 7—May 12, 1979, an NSF Regional Conferenc e wa s held a t th e
University o f Georgi a i n Athens . Th e topic o f th e conferenc e wa s "Specia l divisor s o n alge-
braic curves" , and a t tha t tim e a n informal se t o f lecture notes wit h th e sam e title wa s dis-
tributed. Abou t one-hal f th e materia l in thos e notes contained a n exposition o f result s fro m
the literature, while the othe r par t gav e an account o f recent joint work by Enrico Arbarello,
Maurizio Cornalba, Joe Harris, and myself. In writing up this monograph it wa s decided t o
restrict t o a discussio n o f the ver y elementar y aspect s o f th e theory and an explanation with-
out complet e proof s o f a few unpublishe d result s together wit h som e fro m th e recen t litera -
ture, and then t o publis h a n expande d versio n o f th e remaining content s of the Athens notes
in a more traditional researc h format ; specificall y i n [1] (cf . als o [4] , which contains th e
last sectio n and appendi x fro m th e Athen s notes). Thi s monograph, then, gives an exposition
of the elementar y aspect s o f th e theor y o f specia l divisor s together wit h a n explanation o f
some more advanced result s that ar e not to o technical . A s such, it i s intended t o b e a n in-
troduction t o th e recen t sources , especially [3] , [6] , [7] , [1], and [4] .
As with mos t subjects , one ma y approac h th e theor y o f specia l divisor s fro m severa l
points o f view. Th e one that w e have adopted her e pertain s to Clifford' s theorem , an d ma y
be informally state d a s follows :
The failure of a maximally strong version of Clifford's theorem to hold
imposes nontrivial conditions on the moduli of an algebraic curve.
More precisely, the "failur e o f a maximally stron g version o f Clifford' s theore m t o hold "
will be seen to b e equivalent t o th e failur e o f a certain mappin g n
0
(cf . (1.30)) to be injec -
tive, and in this for m ou r poin t o f vie w on specia l divisor s dates at least t o K . Petri [9] . W e
learned o f Petri's work and o f its relevance t o the Brill-Noethe r proble m fro m th e pape r o f
Arbarello-Sernesi [2] .
This monograph contain s tw o sections , respectively studyin g special divisors using the
Riemann-Roch theore m an d the Jacobia n variety . I n the firs t sectio n w e begin prett y muc h
at groun d zero , so that a reader wh o has onl y passin g familiarity wit h Rieman n surface s o r
algebraic curves may be able to follo w th e discussion . Th e respective subtopic s i n thi s first
section ar e (a) the Riemann-Roc h theorem , (b) Clifford's theore m an d th e pi0-mapping, and
(c) canonical curve s and th e Brill-Noethe r matrix . I n th e secon d sectio n w e assume a little
more, although agai n an attempt ha s been mad e to explain , if not prove , anything. Th e re-
spective subtopics ar e (a) Abel's theorem, (b ) the reappearanc e o f the Brill-Noethe r matri x
1
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