Canonical Resolution of a Quasi-ordinary Surface Singularity

Ban, Chunsheng ; McEwan, Lee J.

Canadian journal of mathematics, Tome 52 (2000) no. 6, pp. 1149-1163

Résumé

We describe the embedded resolution of an irreducible quasi-ordinary surface singularity $\left( V,\,p \right)$ which results from applying the canonical resolution of Bierstone-Milman to $\left( V,\,p \right)$ . We show that this process depends solely on the characteristic pairs of $\left( V,\,p \right)$ , as predicted by Lipman. We describe the process explicitly enough that a resolution graph for $f$ could in principle be obtained by computer using only the characteristic pairs.

Détail

DOI : 10.4153/CJM-2000-048-1

Mots-clés : 14B05, 14J17, 32S05, 32S25, canonical resolution, quasi-ordinary singularity

Bibliographie

[A] [A] Abhyankar, S. S., Resolution of Singularities of Embedded Algebraic Surfaces. Springer Monographs in Math., 2nd Edition, 1998. Google Scholar

[BM1] [BM1] Bierstone, E. and Milman, P., A simple constructive proof of canonical resolution of singularities. Effective Methods in Algebraic Geometry, Progr. Math. 94(1991), 11–30. Google Scholar

[BM2] [BM2] Bierstone, E. and Milman, P., Resolution of singularities, (alg-geom/979028), preprint. Google Scholar

[BM3] [BM3] Bierstone, E. and Milman, P., Canonical Desingularization in Characteristic Zero by Blowing Up theMaximum Strata of a Local Invariant. Invent. Math. 128(1997), 207–302. Google Scholar

[EV] [EV] Encinas, S. and Villamayor, O., Good points and algorithmic resolution of singularities Acta Math. 181(1998), 109–158. Google Scholar

[G] [G] Gau, Y.-N., Embedded topological classification of quasi-ordinary singularities. Mem. Amer. Math. Soc. 74(1988), 109–129. Google Scholar

[H] [H] Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero I, II. Ann. of Math. 79(1964), 109–326. Google Scholar

[L1] [L1] Lipman, J., Quasi-ordinary singularities of embedded surfaces. Ph.D. thesis, Harvard University, 1965. Google Scholar

[L2] [L2] Lipman, J., Quasi-ordinary singularities of surfaces in C3. Singularities, Proc. Symp. Pure Math. 40, Amer. Math. Soc. Providence 1983, Part 2, 161–171. Google Scholar

[L3] [L3] Lipman, J., Topological invariants of quasi-ordinary singularities. Mem. Amer. Math. Soc. 74(1988), 1–107. Google Scholar

[L4] [L4] Lipman, J., Equiresolution and Simultaneous Resolution of Singularities. Proceedings of Tirol Conference on Resolution of Singularities, to appear. Google Scholar

[M] [M] Moh, T. T., Canonical uniformization of hypersurface singularities of characteristic zero. Comm. Algebra 20(1992) 3207–3251. Google Scholar

[O] [O] Orbanz, U., Enbedded resolution of algebraic surfaces after Abhyankar (characteristic), Lecture Notes in Math. 1101, Springer, 1984. Google Scholar

[V1] [V1] Villamayor, O., Constructiveness of Hironaka's resolution. Ann. Sci. École Norm. Sup. (4) 22(1989), 1–32. Google Scholar

[V2] [V2] Villamayor, O., On Equiresolution and a Question of Zariski. preprint. Google Scholar