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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 OX-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?

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I never understand what is precisely meant by "yoga" in mathematics. Of course it will not have a precise meaning, but it still seems to be used in at least a couple of different ways. –  Kevin H. Lin Oct 23 '09 at 19:45
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I usually translate of the semantics "yoga of ..." as "some kind of philosophical direction that you would bet would never solve any concrete problem but then suddenly it does solve lots of concrete problems." –  Ilya Nikokoshev Oct 23 '09 at 20:06
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It's a term Grothendieck liked a lot. Usually it referred to an intuitive description of something, the general feel it has, etc... For example his "yoga of motives" was a conjectural set of ideas about what motives should be like (a "universal" cohomology theory) and what properties you can derive from that. –  Sam Derbyshire Oct 24 '09 at 4:57
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Yes, but I think part of semantics is some unexpectedness. –  Ilya Nikokoshev Oct 24 '09 at 21:12
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I interpret "yoga of..." as "mental exercise": some training one must do (usually with lots of concentration) going mentally from one context/concept/idea to another, to grasp the prescribed/alleged relation between them, or applying an abstract idea to some instances. –  Qfwfq Apr 28 '10 at 9:00
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5 Answers

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?)

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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.

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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.

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In circumstances where properties at a point extend to a neighborhood, there's nothing special about the generic point; a neighborhood of any point is dense. I guess I wanted examples where the generic point is really special. Your finiteness example does the trick. –  Anton Geraschenko Oct 24 '09 at 15:30
    
Good point, didn't think that one through. –  Greg Stevenson Oct 24 '09 at 20:47
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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.

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I guess the 'generic points' the Italians were working with were 'points in general position'. –  J.C. Ottem Nov 17 '10 at 13:24
    
Is not the concept of a generic point capturing the concept of points in general position in a brilliant and precise way? 1) Things holding at the generic point usually hold in an open dense subset, i.e. for all "points of general position". 2) If we consider, e.g., $\mathbb{P}^n_k$, then its generic point is a $k(x_1,\dots, x_n)$-valued points; i.e. you can specialize it to every other point by prescribing values of the $x_i$. So, it is really a "general point". –  Lennart Meier Sep 2 '13 at 18:09
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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.

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I don't understand your last paragraph. A generic section from what to SpecZ should be a rational number? thanks –  Qfwfq Apr 28 '10 at 9:04
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