Geometry of Algebraic Curves, Volume 1

Front Cover
Springer Science & Business Media, Nov 11, 2013 - Mathematics - 387 pages
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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Contents

3 Abels Theorem
3
4 Abelian Varieties and the Theta Function
20
6 A Few Words About Moduli
28
APPENDIX
50
1 Applications of the Discussion About Plane Curves with Nodes
56
Geometric Description
67
4 Determinantal Varieties and Porteous Formula
83
5 A Few Applications and Examples
93
5 First Consequences of the Infinitesimal Study of G C and WC
191
CHAPTER V
203
CHAPTER VI
225
2 Kempfs Generalization of the Riemann Singularity Theorem
239
3 The Torelli Theorem
245
APPENDIX
281
3 Theta Characteristics
287
APPENDIX C
295

CHAPTER III
107
2 Castelnuovos Bound Noethers Theorem and Extremal Curves
113
3 The EnriquesBabbage Theorem and Petris Analysis of the Canonical Ideal
123
CHAPTER IV
153
2 The Universal Divisor and the Poincaré Line Bundles
164
3 The Varieties W C and G C Parametrizing Special Linear Series on
176
4 The Zariski Tangent Spaces to G C and WC
185
CHAPTER VII
304
3 The Connectedness Theorem
311
CHAPTER VIII
330
2 Three Applications of the GrothendieckRiemannRoch Formula
333
5 Diagonals in the Symmetric Product
358
Bibliography 375
374
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