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?