Let $Y$ be a nonsingular subvariety of a normal, Cohen-Macaulay variety $X$. Further, let $\pi:X'\to X$ be the blowup of $X$ along $Y$.

Question: Is there a formula for the Picard group of $X'$ involving the Picard group of $X$ and the exceptional locus of the blowup?

For example when $X$ is smooth it is well known that $\textbf{Pic}\, X'\cong \textbf{Pic}\, X \oplus \mathbb{Z}$ and the isomorphism is induced by the pullback $\pi ^*: \textbf{Pic}\, X \to \textbf{Pic}\, X' $ and the class of the exceptional divisor.

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