Guide for the Reader
List of Symbols
CHAPTER I
Preliminaries
l. Divisors and Line Bundles on Curves
2. The Riemann-Roch and Duality Theorems
3. Abel's Theorem
4. Abelian Varieties and the Theta Function
5. Poincar6's Formula and Riemann's Theorem
6. A Few Words About Moduli
Bibliographical Notes
Exercises
A. Elementary Exercises on Plane Curves
B. Projections
C. Ramification and Plucker Formulas
D. Miscellaneous Exercises on Linear Systems
E. Weierstrass Points
F. Automorphisms
G. Period Matrices
H. Elementary Properties of Abelian Varieties
APPENDIX A
The Riemann-Roch Theorem, Hodge Theorem, and
Adjoint Linear Systems
l. Applications of the Discussion About Plane Curves with Nodes
2. Adjoint Conditions in General
CHAPTER 11
Determinantal Varieties
1. Tangent Cones to Analytic Spaces
2. Generic Determinantal Varieties: Geometric Description
3. The Ideal of a Generic Determinantal Variety
4. Determinantai Varieties and Porteous' Formula
(i) Sylvester's Determinant
(ii) The Top Chern Class of a Tensor Product
(iii) Porteous' Formula
(iv) What Has Been Proved
5. A Few Applications and Examples
Bibliographical Notes
Exercises
A. Symmetric Bilinear Maps
B. Quadrics
C. Applications of Porteous' Formula
D. Chern Numbers of Kernel Bundles
CHAPTER I11
Introduction to Special Divisors
1. Clifford's Theorem and the General Position Theorem
2. Castelnuovo's Bound, Noether's Theorem, and Extremal Curves
3. The Enriques-Babbage Theorem and Petri's Analysis of the Canonical Ideal
Bibliographical Notes
Exercises
A. Symmetric Products of pi
B. Refinements of Clifford's Theorem
C. Complete Intersections
D. Projective Normality (I)
E. Castelnuovo's Bound on k-Normality
F. Intersections of Quadrics
G. Space Curves of Maximum Genus
H. G. Gherardelli's Theorem
I. Extremal Curves
J. Nearly Castelnuovo Curves
K. Castelnuovo's Theorem
L. Secant Planes
CHAPTER IV
The Varieties of Special Linear Series on a Curve
1. The BrilI-Noether Matrix and the Variety C,
2. The Universal Divisor and the Poincar6 Line Bundles
3. The Varieties W[(C) and G(C) Parametrizing Special Linear Series on a
Curve
4. The Zariski Tangent Spaces to G(C) and W(C)
5. First Consequences of the Infinitesimal Study of GC) and WC)
Biographical Notes
Exercises
A. Elementary Exercises on 0
B. An Interesting Identification
C. Tangent Spaces to Wt(C)
D. Mumford's Theorem for.qJ's
E. Martens-Mumford Theorem for Birational Morphisms
F. Linear Series on Some Complete Intersections
G. Keem's Theorems
CHAPTER V
The Basic Results of the Brill-Noether Theory
Bibliographical Notes
Exercises
A. W (C) on a Curve C of Genus 6
B. Embeddings of Small Degree
C. Projective Normality (II)
D. The Difference Map Oa: Cd x Cd -, J(C) (I)
CHAPTER VI
The Geometric Theory of Riemann's Theta Function
1. The Riemann Singularity Theorem
2. Kempf's Generalization of the Riemann Singularity Theorem
3. The Torelli Theorem
4. The Theory of Andreotti and Mayer
Bibliographical Notes
Exercises
A. The Difference Map O (II)
B. Refined Torelli Theorems
C. Translates of WO- 1, Their Intersections, and the Toreili Theorem
D. Prill's Problem
E. Another Proof of the Torelli Theorem
F. Curves of Genus 5
G. Accola's Theorem
H. The Difference Map a (llI)
I. Geometry of the Abelian Sum Map u in Low Genera
APPENDIX B
Theta Characteristics
1. Norm Maps
2. The Weil Pairing
3. Theta Characteristics
4. Quadratic Forms Over Z/2
APPENDIX C
Prym Varieties
Exercises
CHAPTER Vll
The Existence and Connectedness Theorems for PWd(C)
1. Ample Vector Bundles
2. The Existence Theorem
3. The Conneetedness Theorem
4. The Class of W,(C)
5. The Class of C
Bibliographical Notes
Exercises
A. The Connectedness Theorem
B. Analytic Cohomology of Cd, d < 2g - 2
C. Excess Linear Series
CHAPTER VIII
Enumerative Geometry of Curves
1. The Grothendieck-Riemann-Roch Formula
2. Three Applications of the Grothendicck-Riemann-Roch Formula
3. The Secant Plane Formula: Special Cases
4. The General Secant Plane Formula
5. Diagonals in the Symmetric Product
Bibliographical Notes
Exercises
A. Secant Planes to Canonical Curves
B. Weierstrass Pairs
C. Miscellany
D. Push-Pull Formulas for Symmetric Products
E. Reducibility of Wg-l n (Wg-1 + u) (II)
F. Every Curve Has a Base-Point-Free g1- 1
Bibliography
Index
