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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.

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up vote 5 down vote accepted

I think the answer is, almost.

$E$ might have more than one component. Indeed, since you only assume that $Z$ is generically smooth, $E$ might have many different components. $E = \sum E_i$.

You can always write $\beta^* D = \widetilde{D} + \sum_i \alpha_i E_i$, but you can't use the same $\alpha$ for all of them.

Let me give an example. Suppose $X = \mathbb{A}^2$, $Z = V(x \cdot (x,y) \cdot (x,y^2)) = V(x^3, x^2y^2, x^2y, xy^3)$.
Certainly $Z$ is generically smooth and irreducible (its just not reduced, in higher dimensions, I'm think I can rig reduced examples as well). Set $D = \text{Div}(x^3)$. The blow-up of $X$ has two exceptional divisors.
Note $Z$ is a closed subscheme of $D$ as well. By working in higher dimensions, one can rig examples where the support of $Z$ and $D$ are different.

Anyway, one chart of the blow-up looks like $k[x/y, y^2/x]$ with the obvious map to $\text{Spec} k[x,y]$. On that chart, the two exceptional divisors are $E_1 = \text{Div}(x/y)$ and $E_2 =\text{Div}(y^2/x)$.

On this chart, we compute $\beta^*D$. Note $$x^3 = ((x/y)^2 \cdot (y^2/x))^3$$ and so $$\beta^*D = \widetilde{D} + 6 E_1 + 3 E_2.$$

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Thank you for your answer. Hence what I wanted would be valid only if $E$ is irreducible. I think also that find conditions of $Z$ to ensure the irreducibility of $E$ is very complicated :(... bye –  gio Jul 11 '11 at 15:43
gio, another point you can consider is what do you mean by $E = \beta^{-1}(Z)$. Do you mean scheme theoretically, or set theoretically. In particular, does $E$ have the reduced structure? That's how I interpreted your question in my answer. If not, then the thing you wanted is still not true, but there might be a little more hope in some examples. –  Karl Schwede Jul 11 '11 at 17:20
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