This is a follow up to my [previous question](https://mathoverflow.net/questions/68154/strict-transform-of-a-line-bundle), and I have lowered my demands to a situation as follows:


Let $X$ be an algebraic variety, $\mathcal{I}$ a coherent sheaf of ideals and $\mathcal{L}$ a line bundle on $X$. Let $h\_1,\ldots,h\_k\in\mathcal{L}(X)$ be certain global sections with the property that under any local trivialization $\psi:\mathcal{L}|\_U\xrightarrow{\ \sim\ }\mathcal{O}\_U$, we have $\psi\_U(h\_i|\_U)\in\mathcal{I}(U)$. In other words, $ Z(\mathcal{I})\subseteq Z(h\_i)$ for all $i$. 


Let $\beta:Y:=\mathrm{Bl}\_\mathcal{I}(X)\to X$ be the blow-up of $X$ along $\mathcal{I}$. Now, I am looking for a line bundle $\mathcal{M}$ on $Y$ and global sections $g\_0,\ldots,g\_k\in\mathcal{M}(Y)$ such that $Z(g\_i)$ is the strict transform of $Z(h\_i)$. 


From Artie's comment in my [previous question](https://mathoverflow.net/questions/68154/strict-transform-of-a-line-bundle), I assumed that the bundle $\beta^\ast\mathcal{L}\otimes \mathcal{E}^\vee$ should satisfy my demands, where $\mathcal{E}=\bigoplus\_{d\ge 0} \mathcal{I}^{d+1}$ is the exceptional ideal sheaf of the blow-up and $\mathcal{E}^\vee$ its dual.


My question: Is this the $\mathcal{M}$ I am looking for, and if yes, why? I do no see what the corresponding global sections would be. 



Followup; Strict Transform of a Line Bundle
Jesko, you still need more. The condition $Z(\mathscr I)\subseteq Z(h\_i)$ ensures that the strict transform of these sections will be different from their pull-back, but you also need a condition that guarantees that they will be linearly equivalent. 


Here is the point more concretely: Let $D\_i=Z(h\_i)$ and $\sum E\_j=\beta^{-1}(Z(\mathscr I))$. Now $$\beta^\*D\_i \sim \widetilde D\_i+\sum a\_{ij}E\_j$$ where $\widetilde D\_i$ is the strict transform of the divisor $D\_i$ and the $a\_{ij}$ are the appropriate numbers. In order for $\widetilde D\_i\sim \widetilde D\_{i'}$ to be true you need that $a\_{ij}=a\_{i'j}$ for all $j$. In other words, you need some condition that guarantees that the $D\_i$ (in some complicated sense) have the same multiplicity along $Z(\mathscr I)$.


