TITLE: Resolving singularities in one fell swoop
QUESTION [8 upvotes]: From what I understand, resolution of singularities (in characteristic 0) is proved and implemented inductively. You repeatedly blow up your variety along subsets of your singular locus in a way that decreases the "severity" of your singular locus.
At the end of "On the problem of resolution of singularities in positive characteristic" (link), Hauser says:

"From Hironaka’s theorem it follows (at least in characteristic zero) that there does exist another ideal structure on the singular locus of a variety so that the induced blowup with this center resolves the singularities in one single stroke. Formidable!"

So in contrast to the usual inductive way of resolving singularities, there is a way of resolving singularities "all at once."
Question: What does Hauser mean by "another ideal structure" on the singular locus?
The following was my initial thought. If $k$ is a field of characteristic 0 and $X$ is a variety over $k$ with singular locus $Z$, we want some ideal sheaf $\mathcal{I}$ such that $\mathcal{O}_X/\mathcal{I}$ has support $Z$, and such that blowing up $X$ at $\mathcal{I}$ is smooth. Actually constructing $\mathcal{I}$ is presumably difficult, since it should compensate for any difficulties that arise from resolving $Z$.
However, blow-ups are determined by their center, so this initial thought isn't right. For example, there are isolated singular points that require multiple blow-ups to be resolved. How could such a singular point ever be resolved by a single blow-up? Perhaps there's a larger center containing the singular locus that one should blow up?
Edit: As Donu points out, if a singular locus can be resolved, it can be resolved by a single blow-up for entirely formal reasons. So the locus cut out by an ideal that resolves our singular locus in one fell swoop should contain, but need not be equal to, our singular locus. This answers the previous paragraph.
The question then becomes whether the singular locus determines the "resolving ideal" in any tractable way.

REPLY [7 votes]: The reasons for this are rather formal. Any projective birational morphism of varieties is the blow up of some ideal, see chap II, theorem 7.17, of Hartshorne. So although the statement you quote sounds striking, I don't think it is a terribly useful.
Let me add a few more remarks, even though it won't answer your modified question. Blowing up a complicated ideal or closed subscheme would hard understand geometrically. What makes Hironaka, and various other resolution proofs, useful is not just that the resolution exists, but that it can be achieved by blowing up along a succession of smooth centres.