Geometry of Algebraic Curves: Volume I

E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris (auth.)9780387909974, 0387909974

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).

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Table of contents :
Front Matter….Pages i-xvi
Preliminaries….Pages 1-60
Determinantal Varieties….Pages 61-106
Introduction to Special Divisors….Pages 107-152
The Varieties of Special Linear Series on a Curve….Pages 153-202
The Basic Results of the Brill-Noether Theory….Pages 203-224
The Geometric Theory of Riemann’s Theta Function….Pages 225-303
The Existence and Connectedness Theorems for W d r ( C )….Pages 304-329
Enumerative Geometry of Curves….Pages 330-373
Back Matter….Pages 375-387