I'm interested in having especially simple embedded resolutions of a closed subscheme $Z \subset \mathbb{A}^{n}$ in characteristic zero.

Hironaka gives the existence of a smooth variety $M$ and a proper map $p: M \rightarrow \mathbb{A}^{n}$ that is an isomorphism away from $D=p^{-1}(Z)$ and $D$ has simple normal crossings.

I would like to be able to assume that the intersection of two irreducible components of $D$ is either empty or connected, at least when $n={\rm dim} M >2$.

Is this too much to ask for?

I'm interested in having especially simple embedded resolutions of a closed subscheme $Z \subset \mathbb{A}^{n}$ in characteristic zero.

Hironaka gives the existence of a smooth variety $M$ and a proper map $p: M \rightarrow \mathbb{A}^{n}$ that is an isomorphism away from $D=p^{-1}(Z)$ and $D$ has simple normal crossings.

I would like to be able to assume that the intersection of any two irreducible components of $D$ is either empty or connected. If two irreducible components did have disconnected intersection, then perhaps this could be achieved by blowing up $M$ along the connected components of the intersection.

Is this too much to ask for?