What are the most important instances of the "yoga of generic points"? - MathOverflow

What are the most important instances of the "yoga of generic points"?

2009-10-23T19:32:04Z

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 by Ilya Nikokoshev for What are the most important instances of the "yoga of generic points"?

2009-10-23T19:59:09Z

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.

Answer by Charles Siegel for What are the most important instances of the "yoga of generic points"?

2009-10-23T20:00:25Z

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 by Greg Stevenson for What are the most important instances of the "yoga of generic points"?

2009-10-23T23:01:53Z

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 by S. Carnahan for What are the most important instances of the "yoga of generic points"?

2009-10-24T01:34:06Z

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:

asking for something (like a function) to be non-vanishing.
smoothness (from generic flatness).
morphisms having fiber dimension at least n.
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 by roy smith for What are the most important instances of the "yoga of generic points"?

2010-11-17T06:33:33Z

Greg's example from Hartshorne is actually a special case of a more general situation. Under any dominant map of affine varieties, the inverse image of the generic point is the scheme associated to a finitely generated domain over the function field of the target. Hence by Noether's normalization, this inverse image scheme is a finite cover of an affine space over that function field. It follows that over an open set U of the target, the map factors as a finite cover of the projection of U x k^n --> U, where n is the transcendence degree of the field extension defined by the original map. In Hartshorne's exercise of course n = 0. This is the argument for the structure of a dominant morphism in Mumford's red book, I.8, proof of theorem 3.

In my experience the word yoga has been explained to mean "yoke" or "union", from the Sanscrit, and refers to any practice meant to help achieve oneness, or perhaps understanding, as the unknown touched upon above. But my impression is that practicing yoga is more spiritual than intellectual.