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Table of Contents
Intro
Preface
Abridged History of the Theory of Curves and Surfaces
Contents of the Volume
Contents
1 The P12-Theorem: The Classification of Surfaces and Its Historical Development
1.1 Introduction
1.2 Lecture I: The Basic Set Up
1.2.1 First New Concepts Introduced by Enriques
1.2.1.1 Intersection Product
1.2.1.2 The Severi Group and the Neron-Severi Group
1.2.2 The Canonical Divisor and Riemann-Roch for Divisors on Surfaces
1.2.2.1 The Hurwitz Formula
1.2.3 The Arithmetic Genus of a Curve on a Surface
1.2.4 Linear Systems and Morphisms
1.2.5 Exceptional Curves of the First Kind and the Theorem of Castelnuovo-Enriques
1.2.6 Birational Invariants of S and the Albanese Variety
1.2.6.1 Irregular Surfaces and the Albanese Variety
1.2.7 Uniqueness Versus Non Uniqueness of Minimal Models
1.2.7.1 Elementary Transformations of Geometrically Ruled Surfaces
1.2.8 Castelnuovo's Key Theorem
1.2.9 Biregular Invariants of the Minimal Model
1.3 Lecture II: First Important Results for the Classification Theorem of Surfaces
1.3.1 A Basic Tool: Unramified Coverings
1.3.2 Castelnuovo's Theorem on Irregular Ruled Surfaces
1.3.3 Surfaces Fibred Over Curves
1.3.4 Castelnuovo's Criterion of Rationality
1.4 Lecture III: The Classification Theorem
1.4.1 Description of the Surfaces with 12 KS 0 (Case II, P12(S)=1)
1.4.2 Hyperelliptic Surfaces
1.5 Lecture IV: Isotriviality. Central Methods and Ideas in the Proof of the P12-Theorem
1.5.1 Structure of the Proof of the Classification Theorem
1.5.1.1 The Canonical Divisor Formula for Elliptic Fibrations
1.5.1.2 On the Existence of Elliptic Fibrations
1.5.1.3 P12 of Elliptic Fibrations
1.5.2 The Special Case KS nef, KS2=0, pg(S)=0, q(S)=1 and the Crucial Theorem
1.5.3 First Transcendental Proof of Isotriviality for Fibre Genus g = 1.
1.5.3.1 All the Fibres Smooth of Genus g=1
1.5.3.2 g=1 and there are multiple fibres.
1.5.4 Second Transcendental Proof of Isotriviality Using Teichmller Space for Fibre Genus g ?2
1.5.5 Modern Proof of Isotriviality Using Variation of Hodge Structures, and the Theorems of Fujita and Arakelov
1.5.5.1 Fujita's and Arakelov's Theorems
1.5.6 Algebraic Approaches by Castelnuovo-Enriques, Bombieri-Mumford
1.5.6.1 Lemma of Enriques and Mumford mum1
1.6 Appendix: Surfaces with Arithmetic Genus -1, Hyperelliptic Surfaces and Elliptic Surfaces According to Enriques
1.6.1 Analysis of Enriques' Argument
1.6.2 An Explicit Example of Surfaces of Type (2.0,0)
1.7 Some Exercises
1.7.1 Exercise 1 : Exceptional Curves of the First Kind
1.7.2 Exercise 2 : Fibred Surfaces with Fibre Genus g=0
1.7.3 Exercise 3 : Minimal K3 Surfaces, Surfaces with KS 0 (KS is Trivial), q(S) = 0
1.7.4 Exercise 4: Enriques' Construction of Enriques Surfaces
Preface
Abridged History of the Theory of Curves and Surfaces
Contents of the Volume
Contents
1 The P12-Theorem: The Classification of Surfaces and Its Historical Development
1.1 Introduction
1.2 Lecture I: The Basic Set Up
1.2.1 First New Concepts Introduced by Enriques
1.2.1.1 Intersection Product
1.2.1.2 The Severi Group and the Neron-Severi Group
1.2.2 The Canonical Divisor and Riemann-Roch for Divisors on Surfaces
1.2.2.1 The Hurwitz Formula
1.2.3 The Arithmetic Genus of a Curve on a Surface
1.2.4 Linear Systems and Morphisms
1.2.5 Exceptional Curves of the First Kind and the Theorem of Castelnuovo-Enriques
1.2.6 Birational Invariants of S and the Albanese Variety
1.2.6.1 Irregular Surfaces and the Albanese Variety
1.2.7 Uniqueness Versus Non Uniqueness of Minimal Models
1.2.7.1 Elementary Transformations of Geometrically Ruled Surfaces
1.2.8 Castelnuovo's Key Theorem
1.2.9 Biregular Invariants of the Minimal Model
1.3 Lecture II: First Important Results for the Classification Theorem of Surfaces
1.3.1 A Basic Tool: Unramified Coverings
1.3.2 Castelnuovo's Theorem on Irregular Ruled Surfaces
1.3.3 Surfaces Fibred Over Curves
1.3.4 Castelnuovo's Criterion of Rationality
1.4 Lecture III: The Classification Theorem
1.4.1 Description of the Surfaces with 12 KS 0 (Case II, P12(S)=1)
1.4.2 Hyperelliptic Surfaces
1.5 Lecture IV: Isotriviality. Central Methods and Ideas in the Proof of the P12-Theorem
1.5.1 Structure of the Proof of the Classification Theorem
1.5.1.1 The Canonical Divisor Formula for Elliptic Fibrations
1.5.1.2 On the Existence of Elliptic Fibrations
1.5.1.3 P12 of Elliptic Fibrations
1.5.2 The Special Case KS nef, KS2=0, pg(S)=0, q(S)=1 and the Crucial Theorem
1.5.3 First Transcendental Proof of Isotriviality for Fibre Genus g = 1.
1.5.3.1 All the Fibres Smooth of Genus g=1
1.5.3.2 g=1 and there are multiple fibres.
1.5.4 Second Transcendental Proof of Isotriviality Using Teichmller Space for Fibre Genus g ?2
1.5.5 Modern Proof of Isotriviality Using Variation of Hodge Structures, and the Theorems of Fujita and Arakelov
1.5.5.1 Fujita's and Arakelov's Theorems
1.5.6 Algebraic Approaches by Castelnuovo-Enriques, Bombieri-Mumford
1.5.6.1 Lemma of Enriques and Mumford mum1
1.6 Appendix: Surfaces with Arithmetic Genus -1, Hyperelliptic Surfaces and Elliptic Surfaces According to Enriques
1.6.1 Analysis of Enriques' Argument
1.6.2 An Explicit Example of Surfaces of Type (2.0,0)
1.7 Some Exercises
1.7.1 Exercise 1 : Exceptional Curves of the First Kind
1.7.2 Exercise 2 : Fibred Surfaces with Fibre Genus g=0
1.7.3 Exercise 3 : Minimal K3 Surfaces, Surfaces with KS 0 (KS is Trivial), q(S) = 0
1.7.4 Exercise 4: Enriques' Construction of Enriques Surfaces