Toric Varieties (Graduate Studies in Mathematics) by David A. Cox (PDF)

Description

Toric varieties form a beautiful and accessible part of modern algebraic geometry. This book covers the standard topics in toric geometry; a novel feature is that each of the first nine chapters contains an introductory section on the necessary background material in algebraic geometry. Other topics covered include quotient constructions, vanishing theorems, equivariant cohomology, GIT quotients, the secondary fan, and the minimal model program for toric varieties. The subject lends itself to rich examples reflected in the 134 illustrations included in the text. The book also explores connections with commutative algebra and polyhedral geometry, treating both polytopes and their unbounded cousins, polyhedra. There are appendices on the history of toric varieties and the computational tools available to investigate nontrivial examples in toric geometry. Readers of this book should be familiar with the material covered in basic graduate courses in algebra and topology, and to a somewhat lesser degree, complex analysis. In addition, the authors assume that the reader has had some previous experience with algebraic geometry at an advanced undergraduate level. The book will be a useful reference for graduate students and researchers who are interested in algebraic geometry, polyhedral geometry, and toric varieties.

User’s Reviews

Editorial Reviews: Review The book under review is an excellent modern introduction to the subject. It covers both classical results and a large number of topics previously available only in the research literature. The presentation is very explicit, and the material is illustrated by many examples, figures, and exercises. The book combines many advantages of an introductory course, a textbook, a monograph, and an encyclopaedia. It is strongly recommended to a wide range of readers from beginners in algebraic geometry to experts in the area. —- Ivan V. Arzhantsev, Mathematical ReviewsThis masterfully written book will become a standard text on toric varieties, serving both students and researchers. The book’s leisurely pace and wealth of background material makes it perfect for graduate courses on toric varieties or for self-study. Researchers will discover gems throughout the book and will find it to be a valuable resource. —- Sheldon Katz

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐A bit about my background: I’m a math PhD student who started out in number theory but who is considering transitioning to more algebraic geometry. I am familiar with algebraic geometry but not an expert. I really wanted a book that has plenty of instructive examples.It is pretty easy to find the main references for toric varieties: Fulton’s book, Oda’s book, and Danilov’s survey article. This book goes into more detail and does more examples. It is written as a textbook, with lots of exercises. Some of the explanations could be better, like in the first chapter where the explanations are still a bit too terse (for me). Also, there are lectures on MSRI on toric varieties, which are helpful to accompany the book. Sometimes the explanations there differ from the book’s.This book is also instructive for people who don’t have an extensive knowledge of algebraic geometry (but some basic knowledge is required — they refer to some introductory books for further reading). Most chapters have an introductory background section which is required in order to understand the later sections. It is a very useful supplement to an algebraic geometry course because of the extensive discussion of examples. For example, one introductory section discusses line bundles. Then the next section discusses the Cartier divisors on toric varieties and gives an example of line bundles on a variety that are and are not generated by global sections. The whole book is carefully written and reader-friendly. For example, instead of just telling you to look in Hartshorne, it gives you the section and often also the page number.The only downside that will put off some readers is the size of the book. It’s massive. I tend to like books like this, because it means that there are lots of explanations, but some people will prefer a slimmer “lecture notes” style like Fulton’s.Overall, this is a book that grad students who know some algebraic geometry can learn a lot from.

⭐This book is among the best books on the topic of toric varieties.

⭐I purchased this as a gift for our son. He loved it!

⭐This should be the principal reference in toric varieties.

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