meaning of $k$-rational for closed subschemes

Question

Let $X$ be a variety over a field $k$. I know the definition of $k$-rational point: a closed point $x$ is $k$-rational if its residue field $k(x)$ is equal to $k$ (in general, $k(x)$ is only a finite extension of $k$).

Now assume that $Z$ is a closed subscheme of $X$. Is there a notion of being $k$-rational? If so, how is it define?

I'm wondering how important it is to consider $k$-rational subschemes in certain constructions if we want the output to be again defined over $k$.

For instance, I've seen that people compute the motive of a blow-up $Bl_Z X$ in terms of the motives of $X$ and $Z$. They always work in the category of motives over $k$, but it is never said something like "assume $Z$ is $k$-rational". Does the blow-up $Bl_Z X$ always have a natural structure of variety over $k$?

Answer 1

I think there is some confusion in your question; I agree with René that probably the correct analogue of rational point in general is something like a geometrically integral subscheme.

I will however address the question about blow-ups, as this is completely unrelated to the first part of your question.

First, any closed subscheme $Z$ of $X$ is defined over $k$, simply by definition.

To be completely explicit, assume for simplicity that $X = \mathbb{A}_k^n$. Then the coordinate ring of $X$ is $A=k[x_1,\cdots,x_n]$. A closed subscheme $Z$ of $X$ is given by the zero locus of some ideal $I$ of $A$. By definition, this ideal is generated by some polynomials defined over $k$, hence $Z$ is defined over $k$, in whatever sense you would care about.

This whole picture of course generalises by considering instead the relationship between closed subschemes and sheaves of ideals, and looking at affine patches.

Now for the question about blow-ups: if $Z$ is a closed subscheme of $X$ then the blow-up $\mathrm{Bl}_Z X$ is also defined over $k$. This can be seen quite easily by using the explicit construction (given in Hartshorne, say), in terms of sheaves of ideals.

Answer 2

Also related to your question is the notion of field of definition: if $Z_K$ is a closed subscheme of $X_K$ for some field extension $K/k$ (e.g. $K$ an algebraic closure of $k$), then there is a minimal subextension $k'/k$ such that $Z_K$ is defined over $k'$. In case $k'=k$ one probably wishes to say that $Z_K$ is $k$-rational. This and related stuff is studied in EGA 4, part 2, §§ 4.8 and 4.9.