《组合凸性和代数几何》是一部学习凸多面体和多面体集合理论，代数几何和这些领域之间的关系以及著名的环面变量理论的入门书籍。第一部分包括多面体理论，介绍大量线性优化，计算科学领域几何方面的数学背景；第二部分用最基本的方式引进环面变量。

Preface

Introduction

Part 1

Combinatorial Convexity

Ⅰ. Convex Bodies

1.Convex sets

2.Theorems of Radon and Carath6odory

3.Nearest point map and supporting hyperplanes

4.Faces and normal cones

5.Support function and distance function

6.Polar bodies

Ⅱ. Combinatorial theory of polytopes and polyhedral sets

!.The boundary complex of a polyhedral set

2.Polar polytopes and quotient polytopes

3.Special types of polytopes

4.Linear transforms and Gale transforms

5.Matrix representation of transforms

6.Classification of polytopes

Ⅲ Polyhedral spheres

1.Cell complexes

2.Stellar operations

3.The Euler and the Dehn-Sommerville equations

4.Schlegel diagrams, n-diagrams, and polytopality of spheres

5.Embedding problems

6.Shellings

7.Upper bound theorem

Ⅳ. Minkowski sum and mixed volume

1.Minkowsld sum

2.Hausdorff metric

3.Volume and mixed volume

4.Further properties of mixed volumes

5.Alexandrov-Fenchel's inequality

6.Ehrhart's theorem

7.Zonotopes and arrangements of hyperplanes

Ⅴ. Lattice polytopes and fans

1.Lattice cones

2.Dual cones and quotient cones

3.Monoids

4.Fans

5.The combinatorial Picard group

6.Regular stellar operations

7.Classification problems

8.Fano polytopes

Part 2

Algebraic Geometry

Ⅵ. Toric varieties

1.Ideals and affine algebraic sets

2.Affine toric varieties

3.Toric varieties

4.Invariant toric subvarieties

5.The torus action

6.Toric morphisms and fibrations

7.Blowups and blowdowns

8.Resolution of singularities

9.Completeness and compactness

Ⅶ. Sheaves and projective toric varieties

1.Sheaves and divisors

2.Invertible sheaves and the Picard group

3.Projective toric varieties

4.Support functions and line bundles

5.Chow ring

6.Intersection numbers. Hodge inequality

7.Moment map and Morse function

8.Classification theorems. Toric Fano varieties

Ⅷ. Cohomology of toric varieties

1.Basic concepts

2.Cohomology ring of a toric variety

3.Cech cohomology

4.Cohomology of invertible sheaves

5.The Riemann-Roch-Hh-zebruch theorem

Summary: A Dictionary

Appendix

Comments, historical notes, further exercises, research

problems, suggestions for further reading

References

List of Symbols

Index