Combinatorial Convexity and Algebraic Geometry

ISBN-13: 9781461284765 / Angielski / Miękka / 2011 / 374 str.

G. Nter Ewald

Opis

The aim of this book is to provide an introduction for students and nonspecialists to a fascinating relation between combinatorial geometry and algebraic geometry, as it has developed during the last two decades. This relation is known as the theory of toric varieties or sometimes as torus embeddings. Chapters I-IV provide a self-contained introduction to the theory of convex poly- topes and polyhedral sets and can be used independently of any applications to algebraic geometry. Chapter V forms a link between the first and second part of the book. Though its material belongs to combinatorial convexity, its definitions and theorems are motivated by toric varieties. Often they simply translate algebraic geometric facts into combinatorial language. Chapters VI-VIII introduce toric va- rieties in an elementary way, but one which may not, for specialists, be the most elegant. In considering toric varieties, many of the general notions of algebraic geometry occur and they can be dealt with in a concrete way. Therefore, Part 2 of the book may also serve as an introduction to algebraic geometry and preparation for farther reaching texts about this field. The prerequisites for both parts of the book are standard facts in linear algebra (including some facts on rings and fields) and calculus. Assuming those, all proofs in Chapters I-VII are complete with one exception (IV, Theorem 5.1). In Chapter VIII we use a few additional prerequisites with references from appropriate texts.

Recenzje

"... an excellent addition to the literature of this fascinating research field." J. of Computational and Applied Mathematics / Newsletter on Computational and Applied Mathematics

G. Ewald

Combinatorial Convexity and Algebraic Geometry

"An excellent addition to the literature of this fascinating research field."-JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

"For anyone wishing to discover the connections between polytopes and algebraic geometry, this readable and well-organized text can be recommended."-MATHEMATICAL REVIEWS

Spis treści

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.

O autorze

Günter Ewald, geb. 1929, ist emeritierter Professor für Mathematik der Universität Bochum. 1975-1989 war er Mitglied des Präsidiums des Deutschen Evangelischen Kirchentages.