A scheme $X$ over another scheme $S$ is simply a morphism $X \to S$. In particular, if $K$ is a field, a scheme $X$ over $K$ is given by a morphism $X \to \operatorname{Spec} K$. For a long time, I conflated this with the notion of being defined over $K$.
However, the morphisms don’t even go in the right direction: For if $k \subseteq K$ is a subfield, we have a morphism $\operatorname{Spec} K \to \operatorname{Spec} k$ and composing it with our morphisms $X \to \operatorname{Spec} K$ above allows us to consider $X$ as a scheme over $k$ as well, but $X$ should not automatically be defined over every subfield (which intuitively I think of as being given by polynomial equations with coefficients in $k$).
So here is where I am at: The question “Is (a general scheme) $X$ defined over a field $k$?” does not make sense. We can only ask it if $X$ is already given as a scheme over some extension $K$ of $k$. In this case, we say that $X$ is defined over $k$ if there is a scheme $X_k$ over $k$ such that the fiber product $X_k \times_k \operatorname{Spec} K$ is $X$ (or rather, isomorphic to $X$ as a scheme over $k$).
In particular this would entail that a scheme over $k$ is always defined over $k$ (as a scheme over $k$!). This might explain my long-time confusion.
Is this understanding correct?
Best Answer
Here $k$ and $K$ are fields, and a "$k$-scheme" just means a scheme over $k$.
That's right, you don't say that a $K$-scheme $X$ is defined over a subfield $k \subset K$ just by composing the morphisms $X \rightarrow \operatorname{Spec} K \rightarrow \operatorname{Spec} k$. It's more subtle than that. In particular, one reason we don't do this is because for example if $K = \overline{k}$ then $X$ would no longer be of finite type over $k$ even if it is so over $K$. Usually the kind of $k$-schemes people are interested in are varieties, which by any reasonable definintion (there are some minor differences between different authors in the definition of variety) are finite type.
The usual context in which you encounter the language you're talking about is when you have an ambient $k$-variety $\mathscr X$, and $X$ is a subscheme of the $K$-scheme $\mathscr X_K = \mathscr X \times_{\operatorname{Spec}(k)} \operatorname{Spec}(K)$. By composition $X \rightarrow \mathscr X_K \rightarrow \operatorname{Spec}(K)$, $X$ is naturally a $K$-scheme. We say $X$ is defined over $k$ if there is a subscheme $X_0$ of $\mathscr X$ such that $(X_0)_K = X$. If $X_0$ exists, then it is unique.