Re: Fibers question. Algebraic Geometry
From: Axel Vogt (nonail_at_axelvogt.de)
Date: 07/19/04
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Date: Mon, 19 Jul 2004 20:44:06 +0000 (UTC)
Jesus Rogelio Perez Buendia wrote:
>
> HI all,
>
> If X and Y are algebraic sets and V C X x Y is algebraic also.
> And p:X x Y --->X is the projection.
>
> Which conditions do I need to have that all the fibers p^(-1)(a)
> in V has the same dimension? More accurate: When do we have a pont
> a in X such that de fiber of a in V has dimension < than the generic
> fiber?
>
> Thaks for your time!
This is like locally asking for f: V -> X and thus quite general.
At least over the complex numbers you may find answers/sketches
in G. Fischer "Complex analytic Geometry" LNM 538 looking for
semicontinuity in Chap 3, degeneracy and flatness, espec 3.7:
the image of a holomorphic map with constant fibre dimension
is a locally analytic set, that follows from the factorization
Lemma (f writes as finite&surjective, then projection and then
a closed embedding). If X is _irreducible_ this is related to
the openess of f (p.142 ff and false in general). A reasonable
algebraic notion (besides finite) is flatness. An example you
are looking for may be in the exercises of Hartshorne's book
(sorry for not being specific and also for not having much
feeling for char p > 0).
-- use mail for mail not nonail
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