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Let $X$ be a normal variety over $\mathbb{C}$ and $x\in X$ a singular point.

Let $f:Y^{\nu}\to X$ be the normalized blowup at $x\in X$. (i.e. $f$ is a composition of the blowup $Y:=Bl_xX\to X$ and the normalization $Y^{\nu}\to Y$.)

Let $E\subset Y$ be the exceptional divisor of the blowup and $E'$ the pull-back of $E$ to $Y^{\nu}$. Then $-E'$ is $f$-ample, thus $I:=f_*\mathcal{O}_{Y^{\nu}}(-aE')$ generates $\oplus_n f_*\mathcal{O}_{Y^{\nu}}(-naE')$ for some $a>0$.

Q1. Why is $I$ an $m_{X,x}$-primary ideal in $\mathcal{O}_X$?

Q2. What is the blowup of $X$ along $I$? Is it $Y$? $Y^{\nu}$?

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  • $\begingroup$ Consider $0\to f_* \to f_*\mathcal{O}_{Y^\nu}=\mathcal{O}_X\to f_*\mathcal{O}/I$. The right hand sheaf is coherent and supported at $x$. This would be enough to conclude $I$ is $m_x$ primary. $\endgroup$ Nov 2, 2015 at 17:27

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