Let $\mu:X'\rightarrow X$ be a birational morphism of normal complex projective varieties.

Consider the two ideal sheaves $I_1= \mu_*\mathcal{O}_{X'}(-\sum d(E)E)$, $I_2=\mu_*\mathcal{O}_{X'}(-\sum(d(E)+1)E)$, where the $d(E)$'s are non negative integers and the $E$'s are prime divisors.

Suppose $x\in X$ is such that $x\in \mu(E_0)$ for a prime Cartier divisor $E_0$ such that $d(E_0)>0$. Can we say that the stalk ${(I_2)}_x$ is STRICTLY contained in ${(I_1)}_x$?

Thanks for your help.

jumping(ie, when you have strict containment). If your question is false for surfaces, you should be able to counter-examples based on the combinatorics worked out there. $\endgroup$