Question

In algebraic geometry, an irreducible scheme has a point called "the generic point." The justification for this terminology is that under reasonable finiteness hypotheses, a property that is true at the generic point is actually generically true (i.e. is true on a dense open subset).

For example, there is a result called "generic flatness" (EGA IV (2), Theorem 6.9.1). Suppose Y is locally noetherian and integral, with f:X→Y a morphism of finite type and F is a coherent O_X-module. If F is flat over the generic point of Y (a condition which is always satisfied, since anything is flat over a point), then there is a dense open subscheme U⊆Y such that F is flat over U.

I'm sure that there are lots of instances of this "yoga of generic points", but whenever I try to come up with one, it's kind of lame (in my example above, the condition at the generic point is vacuous). What are the main examples of the yoga of generic points?

Answer 1

In some cases, you're looking for open conditions, and you should therefore expect them to look kind of silly when you try to check over points. Others seem to make use of "semicontinuity plus quasi-compact badness" theorems. Standard examples:

1. asking for something (like a function) to be non-vanishing.
2. smoothness (from generic flatness).
3. morphisms having fiber dimension at least n.
4. representability of a moduli problem.

For some of these, we have to be careful calling them open conditions because of non-faithful behavior like empty fibers. We can't always define the empty case to be bad, e.g., the empty scheme is smooth. (Is "smooth + nonempty geometric fibers" an open condition?)

Answer 2

An example (Hartshorne Ch II Ex 3.7) where the condition at the generic point is not vacuous is:

Suppose that f:X -> Y is a dominant finite type morphism of integral schemes such that Y is irreducible and the fibre over the generic point of Y is finite. Then there exists an open dense subscheme U of Y such that f: f^{-1}(U) -> U is finite.

One can also check flatness over a curve by checking whether certain points get mapped to the generic point.

"Generic vanishing" also holds for coherent sheaves in the sense that a torsion sheaf is defined as one which is not supported at the generic point.

Answer 3

I can't give much in the way of details, but I remember reading that one of the major flaws in the Italian school and their work on surfaces was that they kept using "generic points" without definition in a lot of their proofs, and that that hole was patched by the theory of schemes adding exactly the points they needed.

Answer 4

Under some conditions, the morphism or arithmetic schemes (schemes over Spec ZZ) with given section at a generic point admits a true section. It's a topic called (Néron) minimal models and there are nice pictures being drawn :)

Note that a generic section of a morphism to Spec ZZ is a rational number and a true section is an integer number.