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Let $X\subset\mathbb{P}^n$ be an irreducible variety, and let $Sing(X)$ be its singular locus. Let $Y$ be the blow-up of $\mathbb{P}^n$ along $Sing(X)$. Assume that we know that the strict transform of $X$ in $Y$ is smooth.

Now, let $W\subset\mathbb{P}^n$ be a smooth variety such that $Sing(X)\subset W\subset X$, and let $Z$ be the blow-up of $\mathbb{P}^n$ along $W$. Is it true that the strict transform of $X$ in $Z$ is smooth?

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  • $\begingroup$ With your current wording, consider the case when the intersection of $X$ and $W$ is already a Cartier divisor in $X$. Do you want to assume that the intersection of $W$ with $X$ equals $\text{Sing}(X)$? $\endgroup$ Nov 5, 2014 at 0:21
  • $\begingroup$ No, in my case $Sing(X)$ has codimension two in $X$ while $W$ has codimension one. Since $Sing(X)\subset W$ the divisor $W$ in $X$ is not necessarily Cartier. Take for instance a quadric cone in $\mathbb{P}^3$ and a line in it. $\endgroup$
    – user58018
    Nov 5, 2014 at 0:33
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    $\begingroup$ The point of my first comment is that the result cannot possibly hold in the special case that $X\cap W$ is a Cartier divisor in $X$. There may be some special cases where the result does hold if $X\cap W$ is merely a Weil divisor. However, you will need to add some additional hypotheses. $\endgroup$ Nov 5, 2014 at 0:35

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