Real Algebraic Varieties

Abstract

In the first section, we show how real and complex algebraic sets exhibit strikingly different behavior. In the second section, we define real algebraic varieties. In fact, we shall be concerned almost exclusively with affine real algebraic varieties, i.e. real algebraic sets ∜up to a biregular isomorphism∝ The third section concerns the notion of nonsingularity. In addition to recalling some properties of varieties valid over an arbitrary field of characteristic zero, we stress a few special properties of the real case. The fourth section describes important examples of real algebraic varieties: projective spaces and grassmannians. These are affine real algebraic varieties, which explains why, in the real case, there is much less need to leave the affine framework (as compared to the complex case). In the fifth section, we conclude by giving a few useful constructions, such as blowing up and some constructions specific to the real case: the algebraic “Alexandrov compactifi-cation” and blowing down of a subvariety to a point.