Families and Parameter Spaces

  • Joe Harris
Part of the Graduate Texts in Mathematics book series (GTM, volume 133)

Abstract

Next, we will give a definition without much apparent content, but one that is fundamental in much of algebraic geometry. Basically, the situation is that, given a collection {V b } of projective varieties V b ⊂ ℙ n indexed by the points b of a variety B, we want to say what it means for the collection {V b} to “vary algebraically with parameters.” The answer is simple: for any variety B, we define a family of projective varieties in ℙ n with base B to be simply a closed subvariety V of the product B × ℙ n . The fibers V b = (π1)-1(b) of V over points of b are then referred to as the members, or elements of the family; the variety V is called the total space, and the family is said to be parametrized by B. The idea is that if B ⊂ ℙ m is projective, the family V m × ℙ n will be described by a collection of polynomials F α (Z, W) bihomogeneous in the coordinates Z on ℙ m and W on ℙ n , which we may then think of as a collection of polynomials in W whose coefficients are polynomials on B; similarly, if B is affine we may describe V by a collection of polynomials F α (z, W), which we may think of as homogeneous polynomials in the variables W whose coefficients are regular functions on B.

Keywords

Parameter Space Projective Space Algebraic Geometry Projective Variety Rational Section 
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 1992

Authors and Affiliations

  • Joe Harris
    • 1
  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

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