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Let $X$ be a normal projective variety and $D$ be a Weil divisor on $X$ which is $\mathbb{Q}$-Cartier and Cartier in codimension one.

Can we find a projective birational morphism $\pi\colon Y \rightarrow X$ which is a composition of blow-ups at smooth centers of $X$, such that $D_Y$ the strict transform of $D$ on $Y$ is a Cartier divisor, and the centers of this sequence of blow-ups are contained in the locus where $D$ is not Cartier?

I guess that this has something to do with Hironaka principalization theorems, but I don't find a reference.

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    $\begingroup$ Do you really need that each blowing up has a smooth center? In the principalization scheme in Koll\'{a}r's book (inspired by the theorems of many predecessors), the centers are not all smooth. Indeed, if you allow a single blowing up with non-smooth center at the beginning of the sequence, you can principalize $D$. Namely, first blow up the intersection of $D$ with the union $Z$ of all irreducible components of $\text{Sing}(X)$ that are not contained in $D$. After that, apply Hironaka to the open complement of the strict transform of $Z$. $\endgroup$
    – Jason Starr
    Jul 9, 2018 at 21:12

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