Combinatorial Convexity and Algebraic Geometry

1 Combinatorial Convexity.- I. Convex Bodies.- 1. Convex sets.- 2. Theorems of Radon and Carathéodory.- 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.