Toric Varieties


David A. Cox
Amherst College

John B. Little
College of the Holy Cross

Hal Schenck
University of Illinois at Urbana-Champaign

The Book

The study of toric varieties is a wonderful part of algebraic geometry that has deep connections with polyhedral geometry. Our book is an introduction to this rich subject that assumes only a modest knowledge of algebraic geometry. There are elegant theorems, unexpected applications, and marvelous examples that illustrate the scope and power of modern algebraic geometry.

Contents of the Book

The book consists of fifteen chapters:

There are also three appendices:

Downloading the Book

The book is no longer available on-line since it has now been published by the American Mathematical Society.

The Book has been Published

The book has been published by the American Mathematical Society as Volume 124 of their Graduate Studies in Mathematics series. Click here for the AMS page for the book.

Typographical Errors

A pdf file of typographical errors is available for the book Toric Varieties.

Computer Algebra Packages for Toric Varieties

Appendix B of the book deals with computational methods in toric geometry. The two main general-purpose toric packages mentioned in the text are:

Here are two other general-purpose toric packages not discussed in the book that may be of interest:

The book also mentions the computer packages Normaliz, LattE, PALP, Polymake, 4ti2, Gfan, and TOPCOM, and the Sage package polyhedra. These do more specialized tasks in parts of toric geometry (and many other things as well). Here are three additional specialized toric packages not mentioned in the book:

The book also mentions the Macaulay 2 package toriccodes by Nathan Ilten (see reference [153] in the Bibliography). This package is no longer maintained by the author.

Additional References

Here are some relevant references that appeared after publication of the book:

Nonnormal Toric Varieties

While nonnormal toric varieties are defined in Section 3.1 of the book, most of the subsequent text assumes normality. Part of the reason for this is that a nonnormal toric variety need not come from a fan (see Example 3.A.1). However, when a nonnormal toric variety has a torus-invariant affine open cover (automatic in the normal case by Sumihiro's Theorem), then a nicer structure emerges. Here are three references that explore what happens in this case:

Contacting the Authors

You can contact the authors at the following email addresses:

The web site for the book is: