Let $Y$ be an affine, integral, Gorenstein surface. Let $y \in Y$ be a closed point such that there exists a finite, etale morphism $f: X \to Y\backslash \{y\}$ from an integral variety $X$ to the open subvariety $Y \backslash y$. Does there exist an integeral variety $\overline{X}$ containing $X$ as an open dense subscheme and a finite morphism $\overline{f}: \overline{X} \to Y$ restricting to $f$ over $X$ such that the fiber over $y$ consists of exactly one closed point (not necessarily reduced)?
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2$\begingroup$ Yes: first choose any finite morphism $g:Z \to Y$ extending $f$ with $Z$ integral (e.g., by normalizing $Y$ in $X$), and then construct $\overline{X}$ by crushing the finite closed subscheme $g^{-1}(y) \subset Z$ to a single point (so take a suitable fibre product at the level of rings). $\endgroup$– AnonymousCommented Jul 12, 2020 at 3:13
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$\begingroup$ @Anonymous Thank you. Could you put this in the answer. Especially, could you elaborate a little (or give reference) for what you mean by "crushing". Is it something like blowing down? If so, why does a "crushing" always exist? I think the answer lies in what you say "suitable fibre product at the level of rings", but this statement is not clear to me. $\endgroup$– user45397Commented Jul 12, 2020 at 5:55
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1$\begingroup$ For the existence of the crushing (also called "pinching", or Ferrand pushout) see this paper, or this section of the Stacks Project. $\endgroup$– Laurent Moret-BaillyCommented Jul 12, 2020 at 7:19
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1$\begingroup$ In this case the construction is very simple: take $g$ as in Anonymous's comment. Then $\ol{X}=\mathrm{Spec}(\mathscr{A}))$ where $\mathscr{A}\subset g_*\mathscr{O}_Z$ is the subalgebra of sections whose restriction on $g^{-1}(y)$ is constant (i.e. comes from $\kappa(y)$). $\endgroup$– Laurent Moret-BaillyCommented Jul 12, 2020 at 7:29
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$\begingroup$ @LaurentMoret-Bailly and Anonymous; Thank you for the answer $\endgroup$– user45397Commented Jul 12, 2020 at 10:32
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