This book is based on one-semester graduate courses I gave at Michigan in 1994
and 1998, and at Harvard in 1999. A part of the book is borrowed from an earlier
version of my lecture notes which were published by the Seoul National University [22]. The main changes consist of including several chapters on algebraic
invariant theory, simplifying and correcting proofs, and adding more examples
from classical algebraic geometry. The last Lecture of [22], which contains some
applications to construction of moduli spaces, has been omitted. The book is literally intended to be a first course in the subject to motivate a beginner to study
more. A new edition of D. Mumford’s book Geometric Invariant Theory withappendices by J. Fogarty and F. Kirwan [74] as well as a survey article of V. Popov
and E. Vinberg [90] will help the reader to navigate in this broad and old subject
of mathematics. Most of the results and their proofs discussed in the present book
can be found in the literature. We include some of the extensive bibliography of
the subject (with no claim for completeness). The main purpose of this book is
to give a short and self-contained exposition of the main ideas of the theory. The
sole novelty is including many examples illustrating the dependence of the quotient on a linearization of the action as well as including some basic constructions
in toric geometry as examples of torus actions on affine space. We also give many
examples related to classical algebraic geometry. Eachchapter ends witha set of
exercises and bibliographical notes. We assume only minimal prerequisites for
students: a basic knowledge of algebraic geometry covered in the first two chapters of Shafarevich’s book [103] and/or Hartshorne’s book [46], a good knowledge
of multilinear algebra and some rudiments of the theory of linear representations
of groups. Although we often use some of the theory of affine algebraic groups,
the knowledge of the group GLn is enoughfor our purpose.