# An Introduction to the Theory of Special Divisors on Algebraic Curves

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*Phillip A. Griffiths*

A co-publication of the AMS and CBMS

In May, 1979, an NSF Regional Conference was held at the University of
Georgia in Athens. The topic of the conference was “Special divisors on
algebraic curves,”. This monograph gives an exposition of
the elementary aspects of the theory of special divisors together with an
explanation of some more advanced results that are not too technical. As such,
it is intended to be an introduction to recent sources.

As with most subjects, one may approach the theory of special divisors from
several points of view. The one adopted here pertains to Clifford's theorem,
and may be informally stated as follows: The failure of a maximally strong
version of Clifford's theorem to hold imposes nontrivial conditions on the
moduli of an algebraic curve.

This monograph contains two sections, respectively studying special divisors
using the Riemann-Roch theorem and the Jacobian variety. In the first section
the author begins pretty much at ground zero, so that a reader who has only
passing familiarity with Riemann surfaces or algebraic curves may be able to
follow the discussion. The respective subtopics in this first section are (a)
the Riemann-Roch theorem, (b) Clifford's theorem and the
\(\mu_0\)-mapping, and (c) canonical curves and the Brill-Noether
matrix. In the second section he assumes a little more, although again an
attempt has been made to explain, if not prove, anything. The respective
subtopics are (a) Abel's theorem, (b) the reappearance of the Brill-Noether
matrix with applications to the singularities of \(W_d\) and the
Kleiman-Laksov existence proof, (c) special linear systems in low genus.

#### Reviews & Endorsements

All discussion, as easy and elementary as possible, is a successful attempt to give an introduction to an advanced theory, that could also be followed by a reader even only superficially familiar with Riemann surfaces.

-- Ciro Ciliberto, Mathematical Reviews