Hi friends,

Can anybody help me with the following question?

I start with a projective smooth variety $X$ over a field $k$ of a characteristic 0 and $D$ a normal crossing divisor on $X$, with irreducible components $D_i$, $i \in I$.

Now I take a smooth subvariety $Z$ of $X$ of codimension 2 meeting $$D_J:=\bigcap_{j \in J} D_j$$ transversally for all $J \subset I$.

Let $$\varphi: \tilde{X} \to X$$ be the blow-up of $X$ along $Z$. Then $\varphi^\ast D$ is a normal crossing divisor on $\tilde{X}$.

I am trying to relate $\mathcal{O}_{\tilde{X}}(\varphi^\ast D)$ on $\tilde{X}$ to the inverse images of $\mathcal{O}_X(D)$ and $\mathcal{O}_Z(D_Z)$, where $D_Z=D \cap Z$ (which is again a normal crossing divisor on $Z$).

I guess the start point is to look at the exact sequence $$ 0 \to \mathcal{I}_Z \to \mathcal{O}_X \to \mathcal{O}_Z \to 0 $$ and tensor it with $\mathcal{O}_X(D)$ to get

$$ 0 \to \mathcal{I}_Z \otimes \mathcal{O}_X(D) \to \mathcal{O}_X(D) \to \mathcal{O}_Z \otimes \mathcal{O}_X(D) \to 0 $$

The last term is $\mathcal{O}_X(D) _{| Z}=\mathcal{O}_Z(D_Z)$, isn't it?

How to continue?

Thanks for your help!