Projective Algebraic Geometry

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


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


Homogeneous Polynomial Projective Variety Total Degree Projective Line Homogeneous Component 
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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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