Let $X$ be a quasi-projective normal $\mathbb C$-variety and $Y\subset X$ a smooth subvariety such that $X$ is normally flat along $Y$ (i.e. the normal cone of $X$ along $Y$ is flat over $Y$).
Is the blow-up $Bl_Y(X)$ normal?
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2$\begingroup$ What do you mean by normally flat? $\endgroup$– SashaCommented Apr 5, 2021 at 18:42
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$\begingroup$ Thank you for your remark. I have included the definition in the question: it means that the normal cone of $X$ along $Y$ is flat over $Y$. $\endgroup$– pi_1Commented Apr 5, 2021 at 18:58
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1$\begingroup$ That is not true. There are isolated normal surface singularities that are not “absolutely isolated “. $\endgroup$– Jason StarrCommented Apr 5, 2021 at 23:31
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$\begingroup$ Thank you for your counter-example. Do you know, please what would be the answer if moreover the normal cone is assumed to be smooth? $\endgroup$– pi_1Commented Apr 6, 2021 at 8:27
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