Affine Algebraic Geometry

Abstract

The purpose of this chapter is to introduce the reader to the basic definitions, ideas and notations of affine algebraic geometry. Loosely speaking, an affine algebraic set is the zero set of finitely many polynomials in an affine space K ⁿ, where K is an algebraically closed field.¹ Some of the results of this chapter are true and proved over arbitrary algebraically closed fields. This is the reason why we do not consider only the case K = ℂ as we will do from Chapter 3 onwards. Admittedly, our main interest in this book is the behavior of such zero sets in the neighborhood of a point. We will discuss and study the algebraic case first, however, simply because the proofs of the statements in this case usually are easier. This is because one does not have to bother one-selves with convergence questions. Having grasped the main geometric and algebraic ideas in the affine case, the reader is hopefully ready to tackle the local case in Chapter 3. A second reason for doing the affine case, too, is that we need to do some algebraic geometry, in order to classify hypersurface singularities, see Chapter 9.