Projective Algebraic Geometry

  • David Cox
  • John Little
  • Donal O’Shea
Part of the Undergraduate Texts in Mathematics book series (UTM)

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 IPn (k). We will then define projective varieties in IPn (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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • David Cox
    • 1
  • John Little
    • 2
  • Donal O’Shea
    • 3
  1. 1.Department of Mathematics and Computer ScienceAmherst CollegeAmherstUSA
  2. 2.Department of MathematicsCollege of the Holy CrossWorcesterUSA
  3. 3.Department of Mathematics, Statistics, and Computer ScienceMount Holyoke CollegeSouth HadleyUSA

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