Geometry of Algebraic Curves: Volume I

by Enrico Arbarello, Maurizio Cornalba, Phillip A. Griffiths

In recent years there has been enormous activity in the theory of algebraic curves. Many long-standing problems have been solved using the general techniques developed in algebraic geometry during the 1950's and 1960's. Additionally, unexpected and deep connections between algebraic curves and differential equations have been uncovered, and these in turn shed light on other classical problems in curve theory. It seems fair to say that the theory of algebraic curves looks completely different now from how it appeared 15 years ago; in particular, our current state of knowledge repre­ sents a significant advance beyond the legacy left by the classical geometers such as Noether, Castelnuovo, Enriques, and Severi. These books give a presentation of one of the central areas of this recent activity; namely, the study of linear series on both a fixed curve (Volume I) and on a variable curve (Volume II). Our goal is to give a comprehensive and self-contained account of the extrinsic geometry of algebraic curves, which in our opinion constitutes the main geometric core of the recent advances in curve theory. Along the way we shall, of course, discuss appli­ cations of the theory of linear series to a number of classical topics (e.g., the geometry of the Riemann theta divisor) as well as to some of the current research (e.g., the Kodaira dimension of the moduli space of curves).

Genres: Mathematics

403 pages, Hardcover

First published December 20, 1984

Reviews

[Andrew · Rating 3 out of 5]

What is the level of this book? Scheme theory is not essential -- we don't come across anything pathological. Cohomology and sheaf theory are essential prerequisites. We occasionally have to use the Lefschetz hyperplane theorem or Hodge theory. Of course the more mathematics the better, and nothing is *essential* but I would imagine someone with just "the first chapter of Hartshorne" (as the book suggests) would have a difficult time.

One thing I like about this is the consistent level. Chapter 1 is in some respects one of the more difficult chapters simply by the pace it moves, but it also collects the important classical ideas all in one place. The book is well paced, with each subsequent chapter introducing a new idea that extends previous ideas, never venturing too far to abstraction unless it is directly called for. The flavor is part geometric, part combinatorial, and it takes a classical perspective. It was certainly surprising to learn about the sophistication of late 19th and early 20th century mathematics in the Italian school, especially surprising to see the gap between the foundations of Brill-Noether theory and the proof of the Brill-Noether existence theorem and related topics.

[Joe Cole · Rating 5 out of 5]

In conclusion, this book should be an invaluable resource to anyone who wishes to learn about schemes, especially with arithmetic applications in mind. For those inclined towards geometry, an account of schemes from this book coupled with applications from another book would be a good combination.

[Joe Cole · Rating 5 out of 5]

In conclusion, this book should be an invaluable resource to anyone who wishes to learn about schemes, especially with arithmetic applications in mind. For those inclined towards geometry, an account of schemes from this book coupled with applications from another book would be a good combination.

[Joe Cole · Rating 5 out of 5]

this is more readable and more stuff is going in this book on scheme. This book only talks about scheme and does not mention old languages at all. Some important topics are not mentioned sufficiently in this book; so it is better to accompany this book with stack project papers.