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My question is if this can be generalized as well: Let $\beta:\widetilde{X}\mathrel{\mathop:}=\mathop{\mathrm{Bl}}_Z(X)\to X$ be the blow-up of a smooth algebraic variety $X$ along a irreducible generically smooth subscheme $Z$. Let $E\mathrel{\mathop:}=\beta^{-1}(Z)$ be the exceptional divisor. Now, let us assume I have a divisor $D$ on $X$, with $Z$ closed subscheme of $D$ ($D$ smooth and irreducible, if it helps). Then I ask if $\beta^\ast D \sim \widetilde{D} + \alpha E$, for some integer $\alpha$, where $\widetilde{D}$ denotes the strict transform of $D$ and "$\sim$" is linear equivalence.

My question is if this can be generalized as well: Let $\beta:\widetilde{X}\mathrel{\mathop:}=\mathop{\mathrm{Bl}}_Z(X)\to X$ be the blow-up of a smooth algebraic variety $X$ along a irreducible generically smooth subscheme $Z$. Let $E\mathrel{\mathop:}=\beta^{-1}(Z)$ be the exceptional divisor. Now, let us assume I have a divisor $D$ on $X$, with $Z$ closed subscheme of $D$ ($D$ smooth and irreducible, if it helps). Then I ask if $\beta^\ast D \sim \widetilde{D} + \alpha E$, for some integer $\alpha$, where $\widetilde{D}$ denotes the strict transform of $D$ and "$\sim$" is linear equivalence.

My question is if this can be generalized as well: Let $\beta:\widetilde{X}\mathrel{\mathop:}=\mathop{\mathrm{Bl}}_Z(X)\to X$ be the blow-up of a smooth algebraic variety $X$ along a irreducible generically smooth subscheme $Z$. Let $E\mathrel{\mathop:}=\beta^{-1}(Z)$ be the exceptional divisor. Now, let us assume I have a divisor $D$ on $X$, with $Z$ closed subscheme of $D$ ($D$ smooth and irreducible, if it helps). Then I was told that $\beta^\ast D \sim \widetilde{D} + \alpha E$, for some integer $\alpha$, where $\widetilde{D}$ denotes the strict transform of $D$ and "$\sim$" is linear equivalence.

My question is if this can be generalized as well: Let $\beta:\widetilde{X}\mathrel{\mathop:}=\mathop{\mathrm{Bl}}_Z(X)\to X$ be the blow-up of a smooth algebraic variety $X$ along a irreducible generically smooth subscheme $Z$. Let $E\mathrel{\mathop:}=\beta^{-1}(Z)$ be the exceptional divisor. Now, let us assume I have a divisor $D$ on $X$, with $Z$ closed subscheme of $D$ ($D$ smooth and irreducible, if it helps). Then I ask if $\beta^\ast D \sim \widetilde{D} + \alpha E$, for some integer $\alpha$, where $\widetilde{D}$ denotes the strict transform of $D$ and "$\sim$" is linear equivalence.

My question is if this can be generalized as well: Let $\beta:\widetilde{X}\mathrel{\mathop:}=\mathop{\mathrm{Bl}}_Z(X)\to X$ be the blow-up of a smooth algebraic variety $X$ along a irreducible generically smooth subscheme $Z$. Let $E\mathrel{\mathop:}=\beta^{-1}(Z)$ be the exceptional divisor. Now, let us assume I have a divisor $D$ on $X$, with $Z$ closed subscheme of $D$ ($D$ smooth, if it helps). Then I was told that $\beta^\ast D \sim \widetilde{D} + \alpha E$, for some integer $\alpha$, where $\widetilde{D}$ denotes the strict transform of $D$ and "$\sim$" is linear equivalence.

My question is if this can be generalized as well: Let $\beta:\widetilde{X}\mathrel{\mathop:}=\mathop{\mathrm{Bl}}_Z(X)\to X$ be the blow-up of a smooth algebraic variety $X$ along a irreducible generically smooth subscheme $Z$. Let $E\mathrel{\mathop:}=\beta^{-1}(Z)$ be the exceptional divisor. Now, let us assume I have a divisor $D$ on $X$, with $Z$ closed subscheme of $D$ ($D$ smooth and irreducible, if it helps). Then I was told that $\beta^\ast D \sim \widetilde{D} + \alpha E$, for some integer $\alpha$, where $\widetilde{D}$ denotes the strict transform of $D$ and "$\sim$" is linear equivalence.