I'd like to check with my colleagues whether I have correctly understood "embedded resolution of singularities". 


Let $X$ be a nonsingular projective variety over $\mathbf C$ and let $D$ be a "nice" divisor on $X$, say $D$ has strictly normal crossings. (Maybe we could just take $D$ to be a closed subscheme?)


Then, has the following statement been proven? And what is a "good" reference?


There exists a projective birational surjective morphism $\psi:Y\to X$ with $Y$ a nonsingular projective variety over $\mathbf C$ and the inverse image of $D$ in $Y$ a nonsingular projective variety over $\mathbf C$ of codimension one in $Y$?


I'm worried about whether I have correctly understood this statement, or maybe one needs some "normality" conditions on $D$ to assure this "embedded" resolution of singularities.


Also, how does one obtain this embedded resolution of singularities? Can we write down a terminating process which ends with an embedded resolution of singularities?


I have a hard time "believing" the above statement, but I don't know why. If anybody can explain to me that this is not so surprising as a result I would be very thankful.



Embedded resolution of singularities
What embedded resolution does achieve is this: if $Z\subset X$ is a Zariski closed set in a variety, then there is a smooth variety $Y\to X$, obtained from a series of blow ups along smooth centers, such that the preimage of $Z$ is a divisor with normal crossings. However, there is no reason to expect that the inverse of image of $Z$ (or your $D$) in a further blow up will be smooth. (Francesco said this already in a comment, but it bears repeating.) This is already clear in the case where D is a union of two lines in the plane. If you take the preimage in a further blow up, you will get a tree of $\mathbb{P}^1$'s. So your instinct is correct here.


