The closure of a point in a moduli stack

Question

Suppose that $\mathcal{X}$ is an (algebraic, finite type over an algebraically closed field $k$) moduli stack of some geometric objects (e.g. curves, abelian varieties, etc.), so that for a scheme $S$ over $k$ we would have that $\mathcal{X}(S)$ is the groupoid of families of such geometric objects over $S$.

If I have a $k$-point $P$ in the topological space $|\mathcal{X}|$ of $\mathcal{X}$, is there a moduli-theoretic interpretation of what the closure of $P$ in $|\mathcal{X}|$ means? For instance, if I have a family $\mathbb{A}^1 \to \mathcal{X}$ which generically looks like some other $k$-point $Q$ but specializes to $P$, does this mean that $Q$ is in the closure of $P$?

Answer 1

I guess we can say the following, though I would still be interested if someone else could provide a different perspective and/or corrections.

If we have a family of such geometric objects over an integral, finite type $k$-scheme $X$, such that the generic member of this family is given by some $k$-point $Q: \operatorname{Spec} k \to \mathcal{X}$, then any point $P: \operatorname{Spec}k \to X \to \mathcal{X}$ is contained in the closure of $Q$ in $|\mathcal{X}|$.

Proof: Let $\mathcal{Z} \to \mathcal{X}$ be a closed substack of $\mathcal{X}$ with $Q \in |\mathcal{Z}|$ (Stacks Tag 0509). We want to show that $P \in |\mathcal{Z}|$. By the definition of a closed immersion, the morphism $X \times_{\mathcal{X}} \mathcal{Z} \to X$ is a closed immersion of schemes, and we have a surjection of topological spaces $|X \times_{\mathcal{X}} \mathcal{Z}| \to |X| \times_{|\mathcal{X}|} |\mathcal{Z}|$ (Stacks Tag 04XH). Moreover, since $Q \in |\mathcal{Z}|$, we have by considering this surjection that $|X \times_{\mathcal{X}} \mathcal{Z}|$ contains a dense open subset of $|X|$. Since the morphism $|X \times_{\mathcal{X}} \mathcal{Z}| \to |X|$ is closed, it follows by irreducibility that $|X \times_{\mathcal{X}} \mathcal{Z}| = |X|$, and consequently there exists a point $P' \in |X \times_{\mathcal{X}} \mathcal{Z}|$ mapping to $P \in |X|$. Considering the image of $P'$ in $|Z|$ via the projection, we find that $P \in |Z|$, as desired.

Conversely, I would expect that if $P \in \overline{\{Q\}}$, then there exists a family over an $X$ as above, maybe by taking a smooth cover of a reduced closed substack $\mathcal{Z} \to \mathcal{X}$ with $|\mathcal{Z}| = \overline{\{Q\}}$.