Toric Varieties

by

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:

• Chapter 1: Affine Toric Varieties
• Chapter 2: Projective Toric Varieties
• Chapter 3: Normal Toric Varieties
• Chapter 4: Divisors on Toric Varieties
• Chapter 5: Homogeneous Coordinates on Toric Varieties
• Chapter 6: Line Bundles on Toric Varieties
• Chapter 7: Projective Toric Morphisms
• Chapter 8: The Canonical Divisor of a Toric Variety
• Chapter 9: Sheaf Cohomology of Toric Varieties
• Chapter 10: Toric Surfaces
• Chapter 11: Toric Resolutions and Toric Singularities
• Chapter 12: The Topology of Toric Varieties
• Chapter 13: Toric Hirzebruch-Riemann-Roch
• Chapter 14: Toric GIT and the Secondary Fan
• Chapter 15: Geometry of the Secondary Fan

There are also three appendices:

• Appendix A: The History of Toric Varieties
• Appendix B: Computational Methods
• Appendix C: Spectral Sequences

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:

• The Macaulay 2 package NormalToricVarieties by Greg Smith.
• The Sage package ToricVarieties by Volker Braun and Andrey Novoseltsev.

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

• The GAP package toric by David Joyner.
• The Magma package Toric Varieties by Jarosław Buczyński and Alexander Kasprzyk.

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 PALP package mori.x by Andreas Braun and Nils-Ole Walliser. This treats three-dimensional CY hypersurfaces in toric varieties.
The Macaulay 2 package ToricVectorBundles by Nathan Ilten. This constructs equivariant vector bundles on toric varieties.
• The Sage package lattice_polytope by Andrey Novoseltsev. This counts lattice points on faces of lattice polytopes and computes duals and nef partitions.
• The Maple package Computational procedures for weighted projective spaces by Michele Rossi and Lea Terracini. This treats weighted projective spaces from a toric point of view. See also their paper Weighted projective spaces from the toric point of view with computational applications.

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:

• In the preface to the book, we give references for papers that deal with toric varieties over an arbitrary field. Another paper to check is the preprint On Toric Schemes by Fred Rohrer arXiv:1107.2713.
• The preface also gives a reference for toric stacks. Two more papers on this subject by Anton Geraschenko and Matthew Satriano are Toric Stacks I: The Theory of Stacky Fans arXiv:1107.1906 and Toric Stacks II: Intrinsic Characterization of Toric Stacks arXiv:1107.1907.
• In Section 12.4, we give some references for introductions to equivariant cohomology. Another introduction is Introduction to equivariant cohomology in algebraic geometry (IMPANGA 2010) by Dave Anderson arXiv:1112.1421.
• The end of Section 8.3 mentions classification results for reflexive polytopes. People have also studied smooth polytopes. Here are two papers:
  • Few smooth d-polytopes with N lattice points by Tristram Bogart, Christian Haase, Milena Hering, Benjamin Lorenz, Benjamin Nill, Andreas Paffenholz, Francisco Santos, Hal Schenck arXiv:1010.3887.
  • A classification of smooth convex 3-polytopes with at most 16 lattice points by Anders Lundman arXiv:1206.4827.
• In Section 11.2, we discuss various versions of resolutions of singularities. In 2003, Teissier asked if it is possible to perform resolution of resolution of singularities inside an equivariant map of toric varieties. An affirmative answer is given in the paper On a Question of B. Teissier by Jenia Tevelev arXiv:1208.3173.
• A discussion of the toric minimal model program can be found in the paper Birational Geometry of Toric Varieties by Edilaine Ervilha Nobili arXiv:1204.3883.
• A detailed discussion of weighted projective spaces from the toric point of view can be found in the papers two papers Michele Rossi and Lea Terracini: Weighted Projective Spaces from the Toric Point of View with Computational Applications arXiv:1112.1677 and Computational Procedures for Weighted Projective Spaces arXiv:1112.1673.

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:

• Fan is to Monoid as Scheme is to Ring: A Generalization of the Notion of a Fan by Howard Thompson math.AG/0306221.
• Toric Geometry and the Semple-Nash Modification by Pedro González Pérez and Bernard Teissier arXiv:0912.0593.
• Toric Varieties, Monoid Schemes and cdh Descent by Guillermo Cortiñas, Christian Haesemeyer, Mark Walker and Charles Weibel arXiv:1106.1389.
• The Geometry of Toric Schemes by Fred Roher arXiv:1207.0605.

Contacting the Authors

You can contact the authors at the following email addresses:

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