Combinatorial Convexity and Algebraic Geometry
Combinatorial Convexity and Algebraic Geometry
复制标题
DOI：
10.1007/978-1-4612-4044-0
复制
发表时间：
1996-10
期刊：
影响因子：
--
通讯作者：
G. Ewald
中科院分区：
文献类型：
--
作者：
G. Ewald
这本书的目的是为学生和非专业人士提供一个介绍组合几何和代数几何之间的迷人关系，因为它已经在过去二十年的发展。这种关系被称为环面簇理论，有时也被称为环面嵌入。第一章至第四章提供了一个独立的介绍理论凸多面体和多面体集，可以独立使用的任何应用代数几何。第五章是本书第一部分和第二部分之间的联系。虽然它的材料属于组合凸性，但它的定义和定理是由复曲面簇激发的。通常他们只是简单地将代数几何事实转换成组合语言。第六章至第八章介绍了一个基本的方式，但其中一个可能不是，专家，是最优雅的。在考虑复曲面簇，许多一般概念的代数几何发生，他们可以处理在一个具体的方式。因此，书的第2部分也可以作为代数几何学的介绍和关于这个领域的更远的文本的准备。这本书的两个部分的先决条件是线性代数（包括环和域的一些事实）和微积分的标准事实。假设这些，第一章到第七章中的所有证明都是完整的，只有一个例外（IV，定理5.1）。在第八章中，我们使用了一些额外的先决条件，并引用了适当的文本。
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.