The aim of this work is to prove the following theorem: Let \(Y\) be a projective variety of dimension 3 over a field \(k\) such that there exists a regular projective variety \(X\) birationally equivalent to \(Y\). Then, there exists a projective morphism \(\pi :X'\to Y\), where \(X'\) is a regular projective variety and \(\pi\) is an isomorphism above \(Y_{reg}\). – With the help of short references [the author, Duke Math. J. 63, No. 1, 57-64 (1991); the author, J. Giraud and U. Orbanz, “Resolution of surface singularities”, Lect. Notes Math. 1101 (1984; Zbl 0553.14003); and J. Lipman in Algebraic Geometry, Proc. Sympos. Pure Math. 29, Arcata 1974, 187-230 (1975; Zbl 0306.14007)], this theorem leads to a proof of desingularization in dimension 3 and characteristic \(\geq 7\).
The proof proceeds in three steps. In I.1, we see that we can assume that the birational morphism \(X..\to Y\) is defined everywhere. In III, we build two birational projective morphisms \(p:Y\to Y\) and \(\widetilde \pi :\widetilde X\to\widetilde Y\), with \(\widetilde X\) regular (\(\widetilde Y\) is in general not regular) such that \(p\) is an isomorphism above \(Y_{reg}\) and the indetermination locus of \(\widetilde \pi^{-1}\) is the union of two disjoint closed subsets \(F_ 1\) and \(F_ 2\) with \(p(F_ 1)\subset\text{Sing}(Y)\). In II, we see that we can modify \(\widetilde \pi\) and \(\widetilde Y\) to get \(F_ 2=\emptyset\). Then the modified \(\widetilde \pi\circ p\) and \(\widetilde Y\) give the desingularization of \(Y\).