This book is a very organized introduction to the study of constructions that really go back to Isaac Newton, one of these now being called a Newton polygon. In the context of modern algebraic geometry, the constructions take place when dealing with the resolution of singularities of varieties. Given a variety X, this procedure asks for a map from a nonsingular variety Y to X, such that the map is an isomorphism over the nonsingular locus of X. It was the case of a plane curve singularity that was essentially solved by Newton. His techniques were generalized considerably beginning in the 1970's, and resulted in the theory of toric varieties, which is the main subject of this book.

Loosely speaking, a toric variety is a complex algebraic variety which is the partial compactification of an algebraic torus. The algebraic torus acts on a point in the toric variety such that the orbit of the point is an embedded copy of the algebraic torus. Toric varieties are excellent concrete examples of algebraic varieties since they are characterized entirely by a combinatorial object called its fan, which is a collection of convex cones.

This book is an fine introduction to toric varieties. The author does a thorough job of detailing the relevant background in the first half of the book, which deals mostly with convexity and the geometry of lattice polytopes. A very interesting discussion of the Picard group is given in the last few sections of this part. This is one of the best discussions I have seen in the literature on this subject as it gives the reader a very intuitive and concrete view of this group.

The second half covers toric varieties in detail with systems of rational functions on a toric variety studied via sheaf theory. The reader familiar with sheaf theory from general algebraic geometry will see it take on a beautifully concrete form in this book. Readers new to algebraic geometry will appreciate the more abstract approach to sheaf theory if they move on to these more advanced treatments. The author gives many examples of the constructions involved with toric varieties. The cohomology of toric varieties is also treated very nicely, and here again, a reader with a modest background in combinatorial topology will follow the presentation. The physicist reader doing research into mirror symmetry will appreciate this book, as toric varieties serve as a good starting point for the constructions in that area.

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