I'm sure this must be covered somewhere, but all the references I have only treat this in very special cases (mostly when working over fields).

Suppose $f : X\rightarrow S$ is smooth of finite presentation with geometrically connected fibers of dimension 1, and let $g_i : S\rightarrow X$ be finitely many sections. Let $X^\circ := X - \sqcup_i g_i(S)$ and let $p^\circ : Y^\circ\rightarrow X^\circ$ be finite etale.

I would like a procedure to find a branched cover of $X$ which agrees with $p^\circ$ over $X^\circ$, ideally a construction that commutes with (arbitrary) base change.

One way is to consider the normalization $X'$ of $X$ in $Y^\circ$ (though this only commutes with smooth base change), so we get a commutative diagram, which I don't know how to draw here, but the point is that the following two compositions are equal:

$$Y^\circ\stackrel{p^\circ}{\rightarrow} X^\circ\hookrightarrow X$$ and $$Y^\circ\stackrel{h}{\rightarrow} X'\stackrel{\nu}{\rightarrow} X$$

Under the mild assumption that $S$ is Nagata, we know that $\nu$ is finite (http://stacks.math.columbia.edu/tag/03GR), so lets assume $S$ is Nagata.

Further, by Zariski's main theorem (http://stacks.math.columbia.edu/tag/02LQ) , we know that $h$ is an open immersion.

My questions are:

- Intuitively, what does this relative normalization do? I really don't have any intuition for how relative normalization will behave over the $g_i$'s... I guess some more specific questions are:
- If it's possible to extend $p^\circ : Y^\circ\rightarrow X^\circ$ to an etale cover of $X$, will the normalization $\nu$ be etale? (ie, does the normalization "prefer" "unramified completions" of the cover?)
- When will $X'$ be smooth over $S$?
- Is this the "best" way to recover a branched cover from its etale locus? Unfortunately, as Jason Starr pointed out, it does not commute with general base change. Is there a way to do this that
*does*commute with base change?

References would be appreciated.

EDIT: Perhaps I shouldn't have used "reconstruct" in the title, since that implies that there exists a unique/canonical/best candidate for an extension of $p^\circ$ to a finite map over $X$. Basically I'm asking if there are decent alternatives to the normalization, and also for some intuition about how the normalization behaves.

notcompatible with base change. If $S$ is $\text{Spec}\ \mathbb{C}[t]$, if $X$ is $\text{Spec}\ \mathbb{C}[t,x]$, if $X'$ is $\text{Spec}\ \mathbb{C}[t,x,y]/\langle t - (y-x)(y+x) \rangle$, and if $X^o$, $Y^o$ are the corresponding etale loci, then the fiber over $t=0$ of the normalization $X'$ of $X$ in $Y^o$ is different from the normalization of the fiber over $t=0$ of $X$ in the fiber over $t=0$ of $Y^o$. $\endgroup$normalization. I suppose I shouldn't have used the word "reconstruct" since that assumes there is a unique/canonical/best candidate for a finite map $p$ extending $p^\circ$. My basic two questions are - is there a functor that is at least as good as normalization at "extending $p^\circ$?", and failing that, I was hoping for someone to explain the intuition behind how normalization behaves above the possibly ramified locus. $\endgroup$