A theorem (Theorem 8.1) of Lipman Lipman1969 shows that the blow-up of a rational singularity on a normal surface is still normal. Does anyone know under what (extra) condition Lipman's results can be generalized to higher dimensions? To be more precise, let $X$ be a normal variety with only rational singularities and $Z$ be an irreducible subvariety of $X$ with codimension $\geq 3$. Is the blow-up $\mathrm{Bl}_ZX$ normal? If not, under what extra condition, $\mathrm{Bl}_ZX$ will be normal?
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1$\begingroup$ maybe at least you need some good condition on $Z$? Otherwise even if $X$ is smooth the blowup might not be normal. $\endgroup$– Chen JiangJan 21, 2019 at 10:05
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$\begingroup$ I guess so. Do you have an example that the blow-up is not normal? Or do you have an example of good conditions? Thank you! $\endgroup$– Fei YEJan 21, 2019 at 19:04
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1$\begingroup$ Do you really need to blowup a subvariety? Instead of blowing up a radical ideal, I believe that blowing up the integral closure of a high power of that ideal will be normal (ie, it is the normalization of the original blowup). $\endgroup$– Karl SchwedeJan 23, 2019 at 18:32
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$\begingroup$ Hi Karl, thank you! I am interested in the relation between $\mathrm{mult}_ZX$ and the self-intersection of the exceptional divisor $E$ on the blow-up when $Z$ is simply a closed point. I am not sure whether $\mathrm{mult}_xX=E^{\mathrm{\dim X}}$ if we consider the normalized blow-up. $\endgroup$– Fei YEJan 25, 2019 at 13:52
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