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?

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    $\begingroup$ One can see quite easily that such $(W,t,u)$ need not exist: Take $U$ to be a curve and $f,g$ to be any two morphisms whose images intersect in a finite set. $\endgroup$ – ulrich May 5 '12 at 14:33
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    $\begingroup$ Hi Jinhyun, Ulrich is right. Basically, you would need to know that the coincidence locus $\lbrace x\mid f(x)= g(x)\rbrace$ is connected. $\endgroup$ – Donu Arapura May 5 '12 at 16:05
  • $\begingroup$ Thank you. I think I was a bit careless in posting the question. Indeed, I was expecting too much here, and there seem to be easy counter examples. $\endgroup$ – Jinhyun Park May 6 '12 at 14:18

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