This volume presents the basic tools of algebra and analytic geometry, including the Weierestraß Division Theorem, the Nullstellensatz, dimension theory, normalization, and further topics. As applications, fundamental facts of singularity theory are presented.

Chapter 1 discusses the necessary algebra, assuming a basic background in linear algebra and abstract algebra, including some Galois theory. Chapter 2 deals with the basics of affine algebraic geometry up to Hilbert's Nullstellensatz and decomposition into irreducible components. Chapter 3 addresses the corresponding basics for local analytic geometry, assuming knowledge of the theory of holomorphic functions in one variable. Chapter 4 is written from the point of view of local analytic geometry, yet also can apply to affine algebraic geometry.

As application to the general theory, Chapter 5 studies the simplest germs of local analytic spaces: plane curve singularities. Topics include Puiseux expansion, semigroups of curves, and resolutions of plane curve singularities.

The remaining chapters discuss topics not usually found in books on local analytic geometry. Chapter 6 addresses the behavior of numerical invariants of curves in families; Chapter 7, standard bases. Chapter 8 is devoted to approximation theorems and Chapter 9, the classification of simple hypersurface singularities. The concluding Chapter 10 gives as application of Grauert's Approximation Theorem, a proof of the existence of a semi-universal deformation of an isolated singularity.

The authors give full proofs of all statements in the book or present them as exercises with sufficient hints. Prerequisites include basic algebra, analysis, and function theory. The volume can be used by advanced undergraduates and graduate students for course study, seminars, or as a reference source for research papers in algebraic and analytic geometry.

A publication of Vieweg Verlag. The AMS is exclusive distributor in North America. Vieweg Verlag Publications are available worldwide from the AMS outside of Germany, Switzerland, Austria, and Japan.

Readership

Advanced undergraduates, graduate students, and research mathematicians interested in local analytic geometry.

Reviews

"Very well written and the authors, assuming very little background on the side of the reader, manage to cover in less than 400 pages a large amount of beautiful material, presented in a didactical way ... an excellent introduction to the subject and a very appropriate textbook for a graduate course on these matters."

-- Zentralblatt MATH

Table of Contents

• Algebra
• Affine algebraic geometry
• Basics of analytic geometry
• Further development of analytic geometry
• Plane curve singularities
• The principle of conservation of number
• Standard bases
• Approximation theorems
• Classification of simple hypersurface singularities
• Deformations of singularities
• Bibliography
• Index