Let $D,E \subset \mathbb{C}^3$ be prime divisors where $D$ is smooth and $E$ is not necessarily smooth. Assume that $D \cap E$ has SNC support and let
$D \cap E = \bigcup \Gamma_i$ be a decomposition into irreducible components.
(Added) Let $f_1 \colon X_1 \rightarrow \mathbb{C}^3$ be a blow-up along a smooth curve $\Gamma_i$ and $\tilde{D}_1 \subset X_1$ be the strict transform of $D$. $f_1$ induces an isomorphism $g_1 \colon \tilde{D}_1 \rightarrow D$.
There is a SNC curve $\bigcup \Gamma^1_i \subset \tilde{D}_1$. We next blow-up one of $\Gamma^1_i$.
Question Is it possible to make the strict transforms of $D$ and $E$ disjoint by blowing-up smooth curves $\Gamma_i$ or its strict transforms on the strict transforms of $D$ several times?
I know that the blow-up of $\mathbb{C}^3$ along the ideal sheaf $\mathcal{I}_D + \mathcal{I}_E$ makes $D$ and $E$ disjoint. However I want the smooth centre blow-ups.
(My thought which uses Karl Schwede's answer comment)
Let $\pi:Y \rightarrow X$ be a principalization of $I_D + I_E$ as in Karl Schwede's answer.
Let $Z \subset X$ be a finite set which is the union of the images of the smooth points centers. I think that $\pi^{-1}(X \setminus Z) \rightarrow X \setminus Z$ is the blow-ups of smooth curves $\Gamma_i \setminus Z$ or their strict transform curves on the strict transforms of $D$.
Let $\pi': Y' \rightarrow X$ be a composite of blow-ups of the smooth curves $\Gamma_i$ or its strict transform inside strict transforms of $D$ which are in same order as $\pi$. That is, we can define $\pi'$ by forgetting smooth points centers and curve centers which are contained in the inverse image of smooth point centers. $\pi'$ is same as $\pi$ outside $Z$.
Let $\tilde{D}', \tilde{E}' \subset Y'$ be the strict transforms of the original divisors $D, E \subset X$. If $\tilde{D}' \cap \tilde{E}' \neq \emptyset$, then $\tilde{D}' \cap \tilde{E}' \subset \pi'^{-1}(Z) \cap \bigcup \tilde{\Gamma}'_i$ and it is finite points. This is a contradiction since $\tilde{D}', \tilde{E}'$ are Cartier divisors on a smooth 3-fold $Y$. Hence $\tilde{D}' \cap \tilde{E} = \emptyset$.
Sorry for the long sentence, but I think that the idea is simple. I just ignored centers which is concerned with 0-dimensional centers. Are there gaps in this argument?
