This is a follow up to my previous question, 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, 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.