Toric embedded resolutions of quasi-ordinary hypersurface singularities

González Pérez, Pedro D.

Annales de l'Institut Fourier, Tome 53 (2003) no. 6, pp. 1819-1881.

Nous construisons deux procédés de résolution plongée d'un germe de singularité quasi- ordinaire d'hypersurface analytique complexe qui ne dépendent que des monômes caractéristiques associés à une projection quasi-ordinaire du germe. Ce résultat est une solution à l'un des problèmes ouverts posés par Lipman dans Equisingularity and simultaneous resolution of singularities, Resolution of Singularities, Progress in Mathematics No. 181, 2000, 485-503. Dans le premier procédé la singularité est plongée comme hypersurface. Dans le deuxième procédé, qui est inspiré par un travail de Goldin et Teissier pour les germes de courbes planes (voir Resolving singularities of plane analytic branches with one toric morphism, loc. cit., pages 315-340), la singularité est replongée convenablement dans un espace affine de dimension plus grande et nous construisons des résolutions plongées avec un seul morphisme torique. Nous comparons ces deux procédés et nous montrons qu'ils coïncident sous certaines hypothèses.

We build two embedded resolution procedures of a quasi-ordinary singularity of complex analytic hypersurface, by using toric morphisms which depend only on the characteristic monomials associated to a quasi-ordinary projection of the singularity. This result answers an open problem of Lipman in Equisingularity and simultaneous resolution of singularities, Resolution of Singularities, Progress in Mathematics No. 181, 2000, 485- 503. In the first procedure the singularity is embedded as hypersurface. In the second procedure, which is inspired by a work of Goldin and Teissier for plane curves (see Resolving singularities of plane analytic branches with one toric morphism, loc. cit., pages 315-340), we re-embed the singularity in an affine space of bigger dimension in such a way that one toric morphism provides its embedded resolution. We compare both procedures and we show that they coincide under suitable hypothesis.

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