Friday, 23 March 2007

terminology - What does «generic» mean in algebraic geometry?

An irreducible scheme BB has a unique generic point etaeta. The generic fiber of a family XtoBXtoB is the fiber XetaXeta over that special point etaeta.



A general fiber XbXb is a fiber over binBbinB that belongs to some fixed open set UsubsetBUsubsetB. And very general means that bb belongs to VV which is a complement of countably many Zariski closed proper subsets ZiZi of BB.



That is the most common modern terminology. In older (and not so old) books sometimes generic is used where general would be more appropriate.



Added in response to Kevin Lin's comment: In classical alg. geometry, people care about general fibers. The scheme theory provides generic fibers, which are really very convenient to have, since they are so concrete. The way "generic to general" usually works is as follows: You prove that the generic fiber has a property P, and that the property P is constructible. Then P holds for any bb in an open neighborhood of etaeta, that is for a general bb. EGAs contain a long list of properties which are constructible in proper (e.g. projective) families: smoothness, CM, normality, etc., etc.



(And, yes, similar things were discussed in multiple other MO questions. One thing MO seriously lacks is a clear organization of the accumulated knowledge, so that people do not constantly ask and answer variations of the same question.)

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