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.

Here is a heuristic argument that there is nothing to explain:

The probability that $p$ divides the sum of the preceding primes is $1/p$. So the expected number of primes less than $10^9$ with this property is $\sum_{p \leq 10^9} \frac{1}{p}$. Using Mertens' second theorem,
$$\sum_{p \leq 10^9} \frac{1}{p} \approx \log \log 10^9 + M \approx 3.3$$

Here $\log$ is natural log and $M \approx 0.26149$ is Mertens' constant. 

This is an example of the motto "$\log \log x$ goes to infinity but has never been observed to do so". It is quite common for people to look at primes $p$ which divide some quantity $a_p$ and conclude that they are surprisingly rare when, in fact, they are simply growing as $\log \log N$ for the reason above.
