I recently began to study some aspects of varieties with marked points, and I tried to understand their categorical collective behaviors.
Here is one issue that I was not quite able to see immediately, and I wonder if someone could give me appropriate advice. For simplicity, let $(U, p,q), (V, r,s)$ be two connected varieties with two marked points over a field (nice enough, say). Suppose $p \not = q$, $r \not = s $.
Question : Let $f, g : (U, p, q) \to (V, r,s)$ be two morphisms of varieties $U \to V$ that respect the marked points. Then, can we find a connected variety with two marketed points $(W, t, u)$ with $h : (W, t, u) \to (U, p,q)$ that equalizes $f, g$, in other words, $f \circ h = g \circ h$?
Here, connectedness for $W$ is very important: if not, then I can simply take $(W, t, u) = (p \coprod q, p, q)$, and the inclusion map from $W$ to $U$ will does the job.
If there is only one marked point for each variety, say $f, g : (U, p) \to (V, r)$, then the corresponding question is fairly easy. We can take $(W, t) = (p, p)$.
So, this question looks nontrivial when one has more than one marked point.
I guess when the number of marked points increases, there will be fewer morphisms (even none I guess) so that maybe one can have a nice way to answer it, but I do not know what attempts can be made to it.
Does someone have nice suggestions to try?