What is the Genus?

Exploring several of the evolutionary branches of the mathematical notion of genus, this book traces the idea from its prehistory in problems of integration, through algebraic curves and their associated Riemann surfaces, into algebraic surfaces, and finally into higher dimensions.

Table of Contents

1 The yvoc; According to Aristotle

Part I Algebraic Curves

2 Descartes and the New World of Curves

3 Newton and the Classification of Curves

4 When Integrals Hide Curves

5 Jakob Bernoulli and the Construction of Curves

6 Fagnano and the Lemniscate

7 Euler and the Addition of Lemniscatic Integrals

8 Legendre and Elliptic Functions

9 Abel and the New Transcendental Functions

10 A Proof by Abel

11 Abel’s Motivations

12 Cauchy and the Integration Paths

13 Puiseux and the Permutations of Roots

14 Riemann and the Cutting of Surfaces

15 Riemann and the Birational Invariance of Genus

16 The Riemann-Roch Theorem

17 A Reinterpretation of Abel’s Works

18 Jordan and the Topological Classification

19 Clifford and the Number of Holes

20 Clebsch and the Choice of the Term “Genus”

21 Cayley and the Deficiency

22 Noether and the Adjoint Curves

23 Klein, Weyl, and the Notion of an Abstract Surface

24 The Uniformization of Riemann Surfaces

25 The Genus and the Arithmetic of Curves

26 Several Historical Considerations by Weil

27 And More Recently?

Part II Algebraic Surfaces

28 The Beginnings of a Theory of Algebraic Surfaces

29 The Problem of the Singular Locus

30 A Profusion of Genera for Surfaces

31 The Classification of Algebraic Surfaces

32 The Geometric Genus and the Newton Polyhedron

33 Singularities Which Do Not Affect the Genus

34 Hodge’s Topological Interpretation of Genera

35 Comparison of Structures

Part Ill Higher Dimensions

36 Hilbert’s Characteristic Function of a Module

37 Severi and His Genera in Arbitrary Dimension

38 Poincare and Analysis Situs

39 The Homology and Cohomology Theories

40 Elie Cartan and Differential Forms

41 de Rham and His Cohomology

42 Hodge and the Harmonic Forms

43 Weil’s Conjectures

44 Serre and the Riemann-Roch Problem

45 New Ingredients

46 Whitney and the Cohomology of Fibre Bundles

47 Genus Versus Euler-Poincare Characteristic

48 Harnack and Real Algebraic Curves

49 The Riemann-Roch-Hirzebruch Theorem

50 The Riemann-Roch-Grothendieck Theorem