Introduction to Toric Varieties. (AM-131), Volume 131

William Fulton

Published by Princeton University Press 1993

Introduction to Toric Varieties. (AM-131), Volume 131

William Fulton

Volume 131 in the series Annals of Mathematics Studies

https://doi.org/10.1515/9781400882526

Overview
Contents

About this book

Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic geometry notions such as singularities, birational maps, cycles, homology, intersection theory, and Riemann-Roch translate into simple facts about polytopes, toric varieties provide a marvelous source of examples in algebraic geometry. In the other direction, general facts from algebraic geometry have implications for such polytopes, such as to the problem of the number of lattice points they contain. In spite of the fact that toric varieties are very special in the spectrum of all algebraic varieties, they provide a remarkably useful testing ground for general theories.

The aim of this mini-course is to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications. The text concludes with Stanley's theorem characterizing the numbers of simplicies in each dimension in a convex simplicial polytope. Although some general theorems are quoted without proof, the concrete interpretations via simplicial geometry should make the text accessible to beginners in algebraic geometry.

Author / Editor information

William Fulton is Professor of Mathematics at the University of Chicago.

Topics

Frontmatter Publicly Available Download PDF	i
Contents Publicly Available Download PDF	vii
Preface Requires Authentication Unlicensed Licensed Download PDF	ix
Errata Requires Authentication Unlicensed Licensed Download PDF	xii
Chapter 1. Definitions and examples Requires Authentication Unlicensed Licensed Download PDF	1
Chapter 2. Singularities and compactness Requires Authentication Unlicensed Licensed Download PDF	28
Chapter 3. Orbits, topology, and line bundles Requires Authentication Unlicensed Licensed Download PDF	51
Chapter 4. Moment maps and the tangent bundle Requires Authentication Unlicensed Licensed Download PDF	78
Chapter 5. Intersection theory Requires Authentication Unlicensed Licensed Download PDF	96
Notes Requires Authentication Unlicensed Licensed Download PDF	131
References Requires Authentication Unlicensed Licensed Download PDF	149
Index of Notation Requires Authentication Unlicensed Licensed Download PDF	151
Index Requires Authentication Unlicensed Licensed Download PDF	155