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Say I've got a variety X (or a scheme locally of finite type) over an algebraically closed field k. Then closed points of X correspond to k-points of X. (correct?)

Let's define a geometric point of X as a morphism from an algebraically closed field into X. (thus for example the morphism from k[x] to the algebraic closure of the field of fractions of k[x] is a geometric point of the line)

If a (reasonable!) property P holds for all k-points of X does it then hold for all geometric points?

My question comes from moduli stuff. For example, if E is a flat family of sheaves on X parameterised by some base S, such that the fibre of E has some behaviour over all k-points of S, will this behaviour persist on geometric points?

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    thanks for the links David.2011-10-04

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