What is the Genus?

Front Matter....Pages i-xvii
Front Matter....Pages 1-1
Descartes and the New World of Curves....Pages 3-3
Newton and the Classification of Curves....Pages 5-6
When Integrals Hide Curves....Pages 7-8
Jakob Bernoulli and the Construction of Curves....Pages 9-10
Fagnano and the Lemniscate....Pages 11-13
Euler and the Addition of Lemniscatic Integrals....Pages 15-16
Legendre and Elliptic Functions....Pages 17-18
Abel and the New Transcendental Functions....Pages 19-20
A Proof by Abel....Pages 21-22
Abel’s Motivations....Pages 23-24
Cauchy and the Integration Paths....Pages 25-26
Puiseux and the Permutations of Roots....Pages 27-30
Riemann and the Cutting of Surfaces....Pages 31-33
Riemann and the Birational Invariance of Genus....Pages 35-40
The Riemann–Roch Theorem....Pages 41-42
A Reinterpretation of Abel’s Works....Pages 43-44
Jordan and the Topological Classification....Pages 45-49
Clifford and the Number of Holes....Pages 51-52
Clebsch and the Choice of the Term “Genus”....Pages 53-57
Front Matter....Pages 59-61
Cayley and the Deficiency....Pages 3-3
Noether and the Adjoint Curves....Pages 63-64
Klein, Weyl, and the Notion of an Abstract Surface....Pages 65-66
The Uniformization of Riemann Surfaces....Pages 67-68
The Genus and the Arithmetic of Curves....Pages 69-70
Several Historical Considerations by Weil....Pages 71-72
And More Recently?....Pages 73-75
Front Matter....Pages 77-77
The Beginnings of a Theory of Algebraic Surfaces....Pages 79-79
The Problem of the Singular Locus....Pages 81-84
A Profusion of Genera for Surfaces....Pages 85-89
The Classification of Algebraic Surfaces....Pages 91-92
The Geometric Genus and the Newton Polyhedron....Pages 93-95
Singularities Which Do Not Affect the Genus....Pages 97-98
Hodge’s Topological Interpretation of Genera....Pages 99-101
Comparison of Structures....Pages 103-104
Front Matter....Pages 105-106
Hilbert’s Characteristic Function of a Module....Pages 107-107
Severi and His Genera in Arbitrary Dimension....Pages 109-111
Poincaré and Analysis Situs....Pages 113-115
The Homology and Cohomology Theories....Pages 117-120
Elie Cartan and Differential Forms....Pages 121-124
Front Matter....Pages 125-127
de Rham and His Cohomology....Pages 107-107
Hodge and the Harmonic Forms....Pages 129-131
Weil’s Conjectures....Pages 133-136
Serre and the Riemann–Roch Problem....Pages 137-138
New Ingredients....Pages 139-141
Whitney and the Cohomology of Fibre Bundles....Pages 143-145
Genus Versus Euler–Poincaré Characteristic....Pages 147-148
Harnack and Real Algebraic Curves....Pages 149-153
The Riemann–Roch–Hirzebruch Theorem....Pages 155-158
The Riemann–Roch–Grothendieck Theorem....Pages 159-161
Back Matter....Pages 163-166
....Pages 167-186