组合凸性和代数几何

PrefaceIntroductionPart 1Combinatorial ConvexityⅠ. Convex Bodies1.Convex sets2.Theorems of Radon and Carath6odory3.Nearest point map and supporting hyperplanes4.Faces and normal cones5.Support function and distance function6.Polar bodiesⅡ. Combinatorial theory of polytopes and polyhedral sets!.The boundary complex of a polyhedral set2.Polar polytopes and quotient polytopes3.Special types of polytopes4.Linear transforms and Gale transforms5.Matrix representation of transforms6.Classification of polytopesⅢ Polyhedral spheres1.Cell complexes2.Stellar operations3.The Euler and the Dehn-Sommerville equations4.Schlegel diagrams, n-diagrams, and polytopality of spheres5.Embedding problems6.Shellings7.Upper bound theoremⅣ. Minkowski sum and mixed volume1.Minkowsld sum2.Hausdorff metric3.Volume and mixed volume4.Further properties of mixed volumes5.Alexandrov-Fenchels inequality6.Ehrharts theorem7.Zonotopes and arrangements of hyperplanesⅤ. Lattice polytopes and fans1.Lattice cones2.Dual cones and quotient cones3.Monoids4.Fans5.The combinatorial Picard group6.Regular stellar operations7.Classification problems8.Fano polytopesPart 2Algebraic GeometryⅥ. Toric varieties1.Ideals and affine algebraic sets2.Affine toric varieties3.Toric varieties4.Invariant toric subvarieties5.The torus action6.Toric morphisms and fibrations7.Blowups and blowdowns8.Resolution of singularities9.Completeness and compactnessⅦ. Sheaves and projective toric varieties1.Sheaves and divisors2.Invertible sheaves and the Picard group3.Projective toric varieties4.Support functions and line bundles5.Chow ring6.Intersection numbers. Hodge inequality7.Moment map and Morse function8.Classification theorems. Toric Fano varietiesⅧ. Cohomology of toric varieties1.Basic concepts2.Cohomology ring of a toric variety3.Cech cohomology4.Cohomology of invertible sheaves5.The Riemann-Roch-Hh-zebruch theoremSummary: A DictionaryAppendixComments, historical notes, further exercises, researchproblems, suggestions for further readingReferencesList of SymbolsIndex