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Consider $X,Z$ smooth projectives varieties and $Y = Bl_Z(X)$ the blow-up of $X$ with center $Z$. Finally let $E$ be the exceptional divisor of the blow-up. Is is true that $p(Y) - p(E) = p(X) - p(Z)$ where $p$ is the Poincaré polynomial ? Thanks in advance !

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Yes, both sides compute the Poicare polynomial of cohomology with compact supports of $Y \setminus E = X \setminus Z$.

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