Algebraic Curves: Towards Moduli Spaces (Moscow Lectures)

Preface to the Book Series Moscow Lectures......Page 6
Introduction......Page 9
Contents......Page 11
1 Preliminaries......Page 15
 1.1 Complex Projective Spaces......Page 16
 1.3 Two-Dimensional Surfaces......Page 17
 1.4 Gluing Two-Dimensional Surfaces Out of Polygons......Page 18
 1.5 Coverings......Page 20
 1.6 Ramified Coverings......Page 22
 1.7 Riemann–Hurwitz Formula......Page 23
 2.1 Plane Algebraic Curves......Page 26
 2.2 Bézout\'s Theorem and Its Applications......Page 32
 2.3 Rational Parametrization......Page 39
 3.1 A Complex Structure on a Curve......Page 45
 3.2 The Genus of a Smooth Plane Curve......Page 46
 3.3 Hessian and Inflection Points......Page 53
 3.4 Hyperelliptic Curves......Page 54
 3.5 Lifting a Complex Structure......Page 56
 3.6 Quotient Curve......Page 57
 3.7 Meromorphic Functions......Page 58
 4.1 Definition and Examples......Page 62
 4.2 Embeddings and Immersions of Curves......Page 65
 5.1 Projective Duality......Page 69
 5.2 Plücker Formulas for Nonsingular Curves......Page 74
 5.3 Plücker Formulas for Singular Curves......Page 75
 5.4 Newton Polygons......Page 78
 6.1 Automorphisms of the Riemann Sphere......Page 80
 6.2 Mappings of Elliptic Curves......Page 81
 6.3 Moduli of Elliptic Curves......Page 83
 6.4 Lattices and Cubic Curves......Page 86
 6.5 The j-Invariant Revisited......Page 90
 6.6 Automorphisms of Elliptic Curves and Poncelet\'s Closure Theorem......Page 94
 6.7 Automorphisms of Curves of Higher Genus: Hurwitz\'s Theorem......Page 96
 7.1 Tangent and Cotangent Bundles......Page 100
 7.2 How to Define Vector Fields and Differential Forms......Page 103
 7.3 The Dimension of the Space of Holomorphic 1-Forms on a Plane Curve......Page 105
 7.4 Integrating 1-Forms......Page 108
 7.6 Residues and Integrals of Meromorphic 1-Forms......Page 110
 8.1 The Divisor of a Meromorphic Section of a Line Bundle......Page 112
 8.2 The Degree of a Divisor and the Degree of a Bundle......Page 114
 8.3 The Tautological Line Bundle Over the Projective Line......Page 115
 8.4 Recovering a Line Bundle from a Class of Divisors......Page 116
 8.5 Mappings from Curves to Projective Spaces Associated with Line Bundles......Page 117
 8.6 Linear Systems and Mappings Between Curves......Page 118
 9.1 Mittag-Leffler\'s Problem......Page 121
 9.2 The Rational Curve......Page 124
 9.3 Elliptic Curves......Page 125
 9.4 Hyperelliptic Curves and Curves of Genus 2......Page 127
 9.5 Riemann\'s Calculation......Page 128
 9.6 Curves of Genus 3, 4, and 5......Page 129
 10.1 Proof......Page 132
 10.2 Divisors on the Canonical Curve......Page 136
 11.1 Definition of Weierstrass Points......Page 138
 11.3 Weights of Weierstrass Points......Page 140
 11.4 Weierstrass Points and the Finitenessof the Automorphism Group......Page 144
 12.1 Jacobian......Page 146
 12.2 Proof of the Necessity Part......Page 149
 12.3 Proof of the Sufficiency Part: Beginning......Page 150
 12.4 Abelian Differentials of the First, Second, and Third Kind......Page 151
 12.5 Riemann\'s Bilinear Relations......Page 152
 12.6 Completing the Proof of the Sufficiency Part......Page 155
 12.7 Proof of Jacobi\'s Inversion Theorem......Page 157
 12.8 Theta Divisor and Theta Functions......Page 162
 13.1 First Examples......Page 164
 13.2 The Space M1;1......Page 166
 13.3 The Universal Curve Over M1;1......Page 167
 13.4 The Cohomology of the Space M1;1......Page 168
 14.1 Requirements on Moduli Spaces......Page 170
 14.2 A Naive Attempt to Construct a Moduli Space......Page 172
 14.3 Hilbert Polynomial......Page 174
 14.5 Pluricanonical Embeddings......Page 177
 14.6 The Quotient by the Action of the Group of Projective Transformations and Stability......Page 178
 14.7 Moduli Spaces of Curves with Marked Points......Page 181
15 Moduli Spaces of Rational Curves with Marked Points......Page 183
 15.1 Kapranov\'s Construction......Page 184
 15.2 Compactification of Moduli Spaces......Page 186
 15.3 Poincaré Polynomials of Moduli Spaces......Page 188
 15.4 Compactified Spaces: Keel\'s Description of Cohomology......Page 192
 15.5 Compactified Spaces: Kontsevich and Manin\'s Description of Cohomology......Page 197
 16.1 Definition and Examples of Stable Curves......Page 199
 16.2 The Genus of a Nodal Curve......Page 201
 16.3 Degenerations of Smooth Curves......Page 203
 16.4 Compactification of Moduli Spaces by Stable Curves and Pluricanonical Embeddings......Page 204
 17.1 The First Chern Class of a Line Bundle......Page 207
 17.2 Chern Classes of Vector Bundles......Page 208
 17.3 Other Approaches to Defining Chern Classes......Page 212
 17.4 The Genus of a Smooth Plane Curve......Page 215
 17.5 The Genus of a Complete Intersection......Page 216
 17.6 Plücker Formulas......Page 217
 18.1 Rational Curves in the Plane......Page 219
 18.2 Moduli Spaces of Stable Maps......Page 222
 18.3 Quantum Cohomology and the Associativity Equation......Page 224
 19.1 First Term......Page 227
 19.2 Exam Questions......Page 229
 19.3 Second Term......Page 230
 19.4 Exam Questions......Page 232
References......Page 233
Index......Page 235