Combinatorial convexity and algebraic geometry.

*(English)*Zbl 0869.52001
Graduate Texts in Mathematics. 168. New York, NY: Springer. xiv, 372 p. (1996).

The book presents a very good introduction to the theory of convex polytopes (and polyhedral sets) and to algebraic geometry and provides a clear exposition of an important relation between these two fields, namely the theory of toric varieties, or torus embeddings.

The work is divided into two parts. The first one is devoted to combinatorial convexity and starts with the basic notions on convex bodies (Chapter I). Chapter II presents the combinatorial theory of polytopes and polyhedral sets together with a coordinate-free approach to Gale transforms. In Chapter III polyhedral cells and cell complexes are introduced to discuss the topology-oriented part of polytope theory, including Euler and Dehn-Sommerville equations. Chapter IV presents metric results and the fundamental theorems of the Brunn-Minkowski theory. Finally, Chapter V introduces lattice polytopes and fans. The material is treated independently of the application to algebraic geometry and thus can also be suitable for a course on combinatorial convexity.

The second part presents the theory of toric varieties. In Chapter VI affine toric varieties are introduced by lattice cones and the algebra defined by monoids of all lattice points in cones. Toric varieties are then built up by “gluing” affine ones, and important notions like torus action, blow ups, resolution of singularities, completeness and compactness are presented. Chapter VII introduces sheaves and projective toric varieties. Chapter VIII outlines the use of homology and cohomology to study topological properties of toric varieties. A great attempt to simplification and a constant reference to the interactions with combinatorial concepts make general ideas and notions of algebraic geometry accessible to students as well as to those mathematicians, like the reviewer, who are not familiar with this important field.

In the text many examples are given and at the end of each section good selected exercises help the reader to grasp the subject. The work ends with an appendix which collects historical notes, additional exercises, research problems and further references.

The work is divided into two parts. The first one is devoted to combinatorial convexity and starts with the basic notions on convex bodies (Chapter I). Chapter II presents the combinatorial theory of polytopes and polyhedral sets together with a coordinate-free approach to Gale transforms. In Chapter III polyhedral cells and cell complexes are introduced to discuss the topology-oriented part of polytope theory, including Euler and Dehn-Sommerville equations. Chapter IV presents metric results and the fundamental theorems of the Brunn-Minkowski theory. Finally, Chapter V introduces lattice polytopes and fans. The material is treated independently of the application to algebraic geometry and thus can also be suitable for a course on combinatorial convexity.

The second part presents the theory of toric varieties. In Chapter VI affine toric varieties are introduced by lattice cones and the algebra defined by monoids of all lattice points in cones. Toric varieties are then built up by “gluing” affine ones, and important notions like torus action, blow ups, resolution of singularities, completeness and compactness are presented. Chapter VII introduces sheaves and projective toric varieties. Chapter VIII outlines the use of homology and cohomology to study topological properties of toric varieties. A great attempt to simplification and a constant reference to the interactions with combinatorial concepts make general ideas and notions of algebraic geometry accessible to students as well as to those mathematicians, like the reviewer, who are not familiar with this important field.

In the text many examples are given and at the end of each section good selected exercises help the reader to grasp the subject. The work ends with an appendix which collects historical notes, additional exercises, research problems and further references.

Reviewer: C.Peri (Milano)

##### MSC:

52-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to convex and discrete geometry |

52A37 | Other problems of combinatorial convexity |

52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |