The book is an introduction to the theory of convex polytopes and polyhedral sets, to algebraic geometry, and to the connections between these fields, known as the theory of toric varieties.
The book is an introduction to the theory of convex polytopes and polyhedral sets, to algebraic geometry, and to the connections between these fields, known as the theory of toric varieties. The first part of the book covers the theory of polytopes and provides large parts of the mathematical background of linear optimization and of the geometrical aspects in computer science. The second part introduces toric varieties in an elementary way.
1 Combinatorial Convexity.- I. Convex Bodies.- 1. Convex sets.- 2. Theorems of Radon and Caratheodory.- 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.- 1. 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. Minkowski sum.- 2. Hausdorff metric.- 3. Volume and mixed volume.- 4. Further properties of mixed volumes.- 5. Alexandrov-Fenchers 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.- 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. ?ech cohomology.- 4. Cohomology of invertible sheaves.- 5. The Riemann-Roch-Hirzebruch theorem.- Summary: A Dictionary.- Appendix Comments, historical notes, further exercises, research problems, suggestions for further reading.- References.- List of Symbols.
