Algebraic Geometry pp 41-47 | Cite as

# Families and Parameter Spaces

## 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 } = (π_{1})^{-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* ℙ^{ 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

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