Desingularization of Two-Dimensional Schemes

We present a new proof of the existence of a desingularization for any excellent (where surface means two-dimensional reduced noetherian scheme). The problem of resolution of singularities of surfaces has a long history (cf. the expository article [25]). Separate proofs of resolution for arbitrary excellent surfaces were announced by Abhyankar and Hironaka in 1967; to date (1977) full details have not yet been published (but cf. [2], [12], [13], [14] and [15]). Actually Hironaka's results on embedded resolution are stronger than what we shall prove, viz. the following theorem (which nevertheless suffices for many applications). Unless otherwise indicated, all rings in this paper will be commutative and noetherian, and all schemes will be noetherian and reduced. We say that a point z of a scheme Z is regular if the stalk 0,, of the structure sheaf at z is a regular local ring, and singular otherwise; Z is non-singular if all its points are regular.