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 ℙⁿ(k). We will then define projective varieties in ℙⁿ(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.