Algebraic Geometry A First Course (1).pdf

LECTURE 4
Families and Parameter Spaces
Example . Families of Varieties
Next, we will give a defmition without much apparent content, but one that is
fundamental in much of algebraic geometry. Basically, the situation is that, given a
collection {I/b} of projective varieties V, c P’” indexed by the points b of a variety
B, we want to say what it means for the collection {V,} to “vary algebraically with
parameters.” The answer is simple: for any variety B, we define a fumily of projective
varieties in P” with base B to be simply a closed subvariety V of the product
B x P”. The fibers V, = (rcn,)-r(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 c Pm is projective, the
family V c P” x P’ will be described by a collection of polynomials F,(Z, W)
bihomogeneous in the coordinates Z on Pm and Won P”, which we may then think
of as a collection of polynomials in W whose coefftcients are polynomials on B;
similarly, if B is aftine we may describe V by a collection of polynomials F&z, W),
which we may think of as homogeneous polynomials in the variables W whose
coehicients are regular functions on B.
There are many further conditions we can impose on families to insure that
they do indeed vary continuously in various senses; we will discuss some of these
further in Lecture 21.
We should remark here that as a general rule a geometric condition on the
members of a family of varieties V, c P” will determine a constructible, and often
an open or a closed subset of the parameter space B; for example, we use Theorem
to show that for any point p E P” the set of b E B such that p E V, will be a
closed subvariety of B. More generally, we have the following.
Exercise . Let X c P” be any projective variety and {V,> any family of projective
varieties in P” with base B.