Introduction to Algebraic Curves by Phillip A. Griffiths (PDF)

Description

Algebraic curves and compact Riemann surfaces comprise the most developed and arguably the most beautiful portion of algebraic geometry. However, the majority of books written on the subject discuss algebraic curves and compact Riemann surfaces separately, as parts of distinct general theories. Most texts and university courses on curve theory generally conclude with the Riemann-Roch theorem, despite the fact that this theorem is the gateway to some of the most fascinating results in the theory of algebraic curves. This book is based on a six-week series of lectures presented by the author to third- and fourth-year undergraduates and graduate students at Beijing University in 1982. The lectures began with minimal technical requirements (a working knowledge of elementary complex function theory and algebra together with some exposure to topology of compact surfaces) and proceeded directly to the Riemann-Roch and Abel theorems. This book differs from a number of recent books on this subject in that it combines analytic and geometric methods at the outset, so that the reader can grasp the basic results of the subject. Although such modern techniques of sheaf theory, cohomology, and commutative algebra are not covered here, the book provides a solid foundation to proceed to more advanced texts in general algebraic geometry, complex manifolds, and Riemann surfaces, as well as algebraic curves. Containing numerous exercises and two exams, this book would make an excellent introductory text.

User’s Reviews

Editorial Reviews: Review Gives an interesting overview of many of the deep problems in this field of mathematics … a welcome monograph for both students and specialists working on harmonic analysis on Lie groups and homogeneous spaces.” —- Mathematical ReviewsGives an interesting overview of the many problems in this field of mathematics–a welcome monograph for both students and specialists. — G. Olafsson, Mathematical Reviews, 1997Helgason sets a model of style and clarity that has not been matched since Enriques’ “Geometria Projettiva”. This is the kind of mathematics that will live forever. — G-C. Rota, The Bulletin of Mathematical Books, Nov. 1994The style is very fluent and pleasant, conducting the reader at a regular pace … the present book will be a most valuable (and reasonably priced) reference for anyone interested in Radon transforms and analysis on semisimple Lie groups.” — –Bulletin of the AMS

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐perfect!

⭐I wish that I had found this book earlier! This is an extremely readable undergraduate introduction to algebraic curves. I have read Fulton’s “Algebraic Curves” and Miranda’s “Algebraic Curves and Riemann Surfaces”, and while this text covered less material and in less generality, it does so in a very user-friendly and intuitive style. Moreover, it offers the insight of complex algebraic geometry which I personally find to be more intuitive and good motivation for the algebraic approach. For example, consider the proof of Abel’s theorem: first he shows how the problem can be reduced to showing the existence of certain kinds of elementary differentials, which he does using the Normalization theorem and the Riemann Bilinear relations. The whole proof feels extremely natural and reads like a story.More advanced readers will likely find the book too easy and thin. There aren’t very many exercises, though there are a couple of examination problems at the back. $80 seems like a high price for such a small book (disclaimer: I used a library copy). Nonetheless, I gained a great deal of insight and pleasure from this read and I think many could benefit from this concrete, intuitive, and elegant little tome.

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