Families and Parameter Spaces

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 ⊂ ℙn 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 ℙn with base B to be simply a closed subvariety V of the product B × ℙn. 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 ⊂ ℙm is projective, the family V ℙm × ℙn will be described by a collection of polynomials F _α (Z, W) bihomogeneous in the coordinates Z on ℙm and W on ℙn, 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.