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An Introduction to the Theory of Special Divisors on Algebraic Curves
Phillip A. Griffiths
A co-publication of the AMS and CBMS.
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CBMS Regional Conference Series in Mathematics
1980; 25 pp; softcover
Number: 44
ISBN-10: 0-8218-1694-2
ISBN-13: 978-0-8218-1694-3
List Price: US$20
Member Price: US$16
All Individuals: US$16
Order Code: CBMS/44
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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.

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Reviews

"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

Table of Contents

  • The Riemann-Roch theorem and special divisors
  • The Jacobian variety and special divisors
  • Bibliography
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