Ideals, Varieties, and Algorithms pp 349-428 | Cite as

# 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.

## Keywords

Homogeneous Polynomial Projective Variety Total Degree Projective Line Homogeneous Component## Preview

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