Let $X$ be an integral normal scheme over $\mathbb{C}$ with isolated singularity at a closed point $p$. Suppose that $X$ admits a rational resolution $f:Y\to X$ with the exceptional set equals to the fiber $C:=f^{-1}(p)$. It is known that $\mathcal{O}_{X,p}$ is Cohen-Macaulay. Now assume more that $f$ is also small (i.e. the fiber $C$ is of codimension at least $2$), then is it necessary that $\mathcal{O}_{X,p}$ is Gorenstein?
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No. For instance, the cone over $\mathbb{P}^m \times \mathbb{P}^n$ has a small resolution for all $m$ and $n$, but the singularity is Gorenstein only when $m = n$.
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$\begingroup$ If we assume more that the local ring $\mathcal{O}_{X,p}$ is complete, then is it Gorenstein? $\endgroup$ Commented Dec 10, 2018 at 13:25
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$\begingroup$ A noetherian local ring is Gorenstein if and only if its completion is so. $\endgroup$– abxCommented Dec 10, 2018 at 15:01
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$\begingroup$ Is the formal fiber of $f$ at $p$ is still smooth? $\endgroup$ Commented Dec 10, 2018 at 17:29
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$\begingroup$ In this example, is it a rational singularity? $\endgroup$ Commented Dec 11, 2018 at 2:20