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Introduction

I. Linear Sections of an Algebraic Variety

1. Hyperplane Sections of a Non-singular Variety

2. A Family of Linear Sections of W

3. The Fibring of a Variety Defined over the Complex Numbers

4. Homology Groups Related to V(K)

II. The Singular Sections

1. Statement of the Results

2. Proof of Theorem 11

III. A Pencil of Hyperplane Sections

1. The Choice of a Pencil

2. Notation

3. Reduction to Local Theorems

IV. Lefschetz's First and Second Theorems

1. Lefschetz's First Main Theorem

2. Statement of Lefschetz's Second Main Theorem

3. Sketch Proof of Theorem 19

4. Some Immediate Consequences

V. Proof of Lefschetz's Second Theorem

1. Deformation Theorems

2. Some Remarks on Theorem 19

3. Formal Verification of Theorem 19; The Vanishing Cycle

4. Proof of Theorem 19, Parts (1) and (2)

5. Proof of Theorem 19, Part (3)

VI. The Poincaré Formula

1. The Automorphisms Ti

2. Explicit Calculation of T

3. The Formula Τ(γ)=γ-(γ.δ)δ

VII. The Poincaré Formula; Details of Proof

1. Clockwise and Anti-clockwise Isomorphisms

2. A special Representative for §

3. Proof of Theorem 32

4. Proof of Theorem 34

VIII. Invariant Cycles and Relative Cycles

1. Summary of Results Assumed

2. The Intersection and Locus Operators

3. Direct Decomposition for Hr-1(V0,P)

4. Direct Decomposition of Hr-1(V0)

5. Proofs of Theorems 41 and 42

References

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1st Edition - January 1, 1958

Author: Andrew H. Wallace

Editor: I. N. Sneddon

Language: EnglisheBook ISBN:

9 7 8 - 1 - 4 8 3 1 - 5 2 6 0 - 8

Homology Theory on Algebraic Varieties, Volume 6 deals with the principles of homology theory in algebraic geometry and includes the main theorems first formulated by Lefschetz,… Read more

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Homology Theory on Algebraic Varieties, Volume 6 deals with the principles of homology theory in algebraic geometry and includes the main theorems first formulated by Lefschetz, one of which is interpreted in terms of relative homology and another concerns the Poincaré formula. The actual details of the proofs of these theorems are introduced by geometrical descriptions, sometimes aided with diagrams. This book is comprised of eight chapters and begins with a discussion on linear sections of an algebraic variety, with emphasis on the fibring of a variety defined over the complex numbers. The next two chapters focus on singular sections and hyperplane sections, focusing on the choice of a pencil in the latter case. The reader is then introduced to Lefschetz's first and second theorems, together with their corresponding proofs. The Poincaré formula and its proof are also presented, with particular reference to clockwise and anti-clockwise isomorphisms. The final chapter is devoted to invariant cycles and relative cycles. This volume will be of interest to students, teachers, and practitioners of pure and applied mathematics.

Introduction

I. Linear Sections of an Algebraic Variety

1. Hyperplane Sections of a Non-singular Variety

2. A Family of Linear Sections of W

3. The Fibring of a Variety Defined over the Complex Numbers

4. Homology Groups Related to V(K)

II. The Singular Sections

1. Statement of the Results

2. Proof of Theorem 11

III. A Pencil of Hyperplane Sections

1. The Choice of a Pencil

2. Notation

3. Reduction to Local Theorems

IV. Lefschetz's First and Second Theorems

1. Lefschetz's First Main Theorem

2. Statement of Lefschetz's Second Main Theorem

3. Sketch Proof of Theorem 19

4. Some Immediate Consequences

V. Proof of Lefschetz's Second Theorem

1. Deformation Theorems

2. Some Remarks on Theorem 19

3. Formal Verification of Theorem 19; The Vanishing Cycle

4. Proof of Theorem 19, Parts (1) and (2)

5. Proof of Theorem 19, Part (3)

VI. The Poincaré Formula

1. The Automorphisms Ti

2. Explicit Calculation of T

3. The Formula Τ(γ)=γ-(γ.δ)δ

VII. The Poincaré Formula; Details of Proof

1. Clockwise and Anti-clockwise Isomorphisms

2. A special Representative for §

3. Proof of Theorem 32

4. Proof of Theorem 34

VIII. Invariant Cycles and Relative Cycles

1. Summary of Results Assumed

2. The Intersection and Locus Operators

3. Direct Decomposition for Hr-1(V0,P)

4. Direct Decomposition of Hr-1(V0)

5. Proofs of Theorems 41 and 42

References

- No. of pages: 124
- Language: English
- Edition: 1
- Published: January 1, 1958
- Imprint: Pergamon
- eBook ISBN: 9781483152608

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