Projective Algebraic Geometry

Abstract

So far, all of the varieties we have studied have been subsets of affine space k ⁿ. In this chapter, we will enlarge k ⁿ by adding certain “points at ∞” to create n-dimensional projective space ℙ ⁿ(k). We will then define projective varieties in ℙ ⁿ(k) and study the projective version of the algebra-geometry correspondence. The relation between affine and projective varieties will be considered in §4; in §5, we will study elimination theory from a projective point of view. By working in projective space, we will get a much better understanding of the Extension Theorem from Chapter 3. The chapter will end with a discussion of the geometry of quadric hypersurfaces and an introduction to Bezout’s Theorem.