Local monomialization. (Q2717171)

The article gives a short introduction to the problems of local uniformization and local monomialization in their interaction with the general problem of resolution of singularities. Some details are sketched, with references to recent and forthcoming papers of the author. It starts with classical results of Zariski (his fundamental theorem on local uniformization, the theorem on resolution of surface-singularities), illustrates first difficulties, mentions Abhyankar's results on resolution in dimension \(\leq 3\) and Hironaka's famous theorem on the existence of a resolution over a field of characteristic 0. The second part of the paper explains the problem of monomialization (which in general does not have a solution in positive characteristics). Theorem 3.1 [proved in the author's book ``Local monomialization and factorization of morphisms'', Astérisque No. 260 (Paris: Société Mathématique de France) (1999; Zbl 0941.14001)] gives an affirmative answer to a question of Abhyankar on simultaneous resolution along a valuation in characteristic 0. The final section is devoted to several results on monomialization of morphisms. Together with an overview of the proof, the author announces the existence of a global monomialization of a proper dominant morphism from a 3-fold to a surface (in characteristic 0).NEWLINENEWLINEFor the entire collection see [Zbl 0960.00042].