Affine Algebraic Geometry

The theory of toric varieties (also called torus embeddings) describes a fascinating interplay between algebraic geometry and the geometry of convex figures in real affine spaces. This book is a unified up-to-date survey of the various results and interesting applications found since toric varieties were introduced in the early 1970's.

Shafarevich Basic Algebraic Geometry 2 The second edition of Shafarevich's introduction to algebraic geometry is in two volumes. The second volume covers schemes and complex manifolds, generalisations in two different directions of the affine and projective varieties that form the material of the first volume. Two notable additions in this second edition are the section on moduli spaces and representable functors, motivated by a discussion of the Hilbert scheme, and the section on Kähler geometry. The book ends with a historical sketch discussing the origins of algebraic geometry.

This book treats the general theory of Poisson structures and integrable systems on affine varieties in a systematic way. Special attention is drawn to algebraic completely integrable systems. Several integrable systems are constructed and studied in detail and a few applications of integrable systems to algebraic geometry are worked out. In the second edition some of the concepts in Poisson geometry are clarified by introducting Poisson cohomology; the Mumford systems are constructed from the algebra of pseudo-differential operators, which clarifies their origin; a new explanation of the multi Hamiltonian structure of the Mumford systems is given by using the loop algebra of sl(2); and finally Goedesic flow on SO(4) is added to illustrate the linearizatin algorith and to give another application of integrable systems to algebraic geometry.