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?
Best Answer
I'm not exactly sure what you want (see Sasha's question above) but algorithms for embedded resolution of singularities might do what you want (for example, see the recent book of Kollár, various papers by Villamayor and co-authors, or Wlodarczyk)
Indeed, do a log resolution (ie, principalization) of the ideal $I_D + I_E$. What this will produce is a proper birational map $$\pi : Y \to X = \mathbb{C}^3$$ such that
You don't need the SNC support at all, although this will make things much easier.
Of course, I don't quite know what you mean by blowing up the support of the $\Gamma_i$ several times.
In the SNC case: What the resolution algorithms would start with is blowing up the strata of the $D \cap E$, in other words, it first will blow up point where the curves of $D \cap E$ intersect. After doing that, it will next blow up the the strict transforms of irreducible components of $D \cap E$ (ie, the curves). EDIT: It might blow up some more points again at various steps. It might also blow up other curves lying over, but themselves are not the $\Gamma_i$ at a particular step. The reason I say this is because blowing up the $I_D + I_E$ doesn't actually know about the divisors $D$ and $E$ and won't keep track of their strict transforms...
Is this ok for what you have in mind? Or do you want to avoid blowing up the points too?