Author | Ewald, Gรผnter. author |
---|---|

Title | Combinatorial Convexity and Algebraic Geometry [electronic resource] / by Gรผnter Ewald |

Imprint | New York, NY : Springer New York, 1996 |

Connect to | http://dx.doi.org/10.1007/978-1-4612-4044-0 |

Descript | XIV, 374 p. online resource |

SUMMARY

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 polyยญ 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 vaยญ 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

CONTENT

1 Combinatorial Convexity -- I. Convex Bodies -- II. Combinatorial theory of polytopes and polyhedral sets -- III. Polyhedral spheres -- IV. Minkowski sum and mixed volume -- V. Lattice polytopes and fans -- 2 Algebraic Geometry -- VI. Toric varieties -- VII. Sheaves and projective toric varieties -- VIII. Cohomology of toric varieties -- Appendix Comments, historical notes, further exercises, research problems, suggestions for further reading -- References -- List of Symbols

Mathematics
Algebraic geometry
Combinatorics
Mathematics
Combinatorics
Algebraic Geometry