# Combinatorial Convexity and Algebraic Geometry

• Günter Ewald
Textbook

Part of the Graduate Texts in Mathematics book series (GTM, volume 168)

1. Front Matter
Pages i-xiv
2. ### Combinatorial Convexity

1. Front Matter
Pages 1-1
2. Günter Ewald
Pages 3-27
3. Günter Ewald
Pages 29-63
4. Günter Ewald
Pages 65-101
5. Günter Ewald
Pages 103-141
6. Günter Ewald
Pages 143-196
3. ### Algebraic Geometry

1. Front Matter
Pages 197-197
2. Günter Ewald
Pages 199-258
3. Günter Ewald
Pages 259-305
4. Günter Ewald
Pages 307-330
4. Back Matter
Pages 331-374

### Introduction

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.

### Keywords

Dimension Grad Lattice algebraic geometry combinatorial geometry

#### Authors and affiliations

• Günter Ewald
• 1
1. 1.Fakultät für MathematikRuhr-Universität BochumBochumGermany

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4612-4044-0
• Copyright Information Springer-Verlag New York 1996
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-1-4612-8476-5
• Online ISBN 978-1-4612-4044-0
• Series Print ISSN 0072-5285
• Buy this book on publisher's site