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Let $P_1$ be a projective variety over $\mathbb{C}$, and $Y_1 \subseteq P_1$ be a codimension $1$ (irreducible) subvariety. Suppose there is a blowdown morphism $Y_1 \to Y_2$.

Then can I find a projective variety $P_2$ which contains $Y_2$ and birational to $P_1$?

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