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So I was reading about the factorization result that any birational map between smooth varieties is composition of blow-ups and blow-downs with smooth centers. It is apparently true only in ...

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 ...