Toric embedded resolutions of quasi-ordinary hypersurface singularities
Annales de l'Institut Fourier, Volume 53 (2003) no. 6, pp. 1819-1881.

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.

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.

DOI: 10.5802/aif.1993
Classification: 32S15,  32S45,  14M25,  14E15
Keywords: singularities, embedded resolution, discriminant, topological type
González Pérez, Pedro D. 1

1 Université Paris VII, Institut de Mathématiques, UMR CNRS 7586, Équipe Géométrie et et Dynamique, Case 7012, 2 place Jussieu, 75251 Paris Cedex 05 (France)
@article{AIF_2003__53_6_1819_0,
     author = {Gonz\'alez P\'erez, Pedro D.},
     title = {Toric embedded resolutions of quasi-ordinary hypersurface singularities},
     journal = {Annales de l'Institut Fourier},
     pages = {1819--1881},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {53},
     number = {6},
     year = {2003},
     doi = {10.5802/aif.1993},
     zbl = {1052.32024},
     mrnumber = {2038781},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1993/}
}
TY  - JOUR
AU  - González Pérez, Pedro D.
TI  - Toric embedded resolutions of quasi-ordinary hypersurface singularities
JO  - Annales de l'Institut Fourier
PY  - 2003
DA  - 2003///
SP  - 1819
EP  - 1881
VL  - 53
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.1993/
UR  - https://zbmath.org/?q=an%3A1052.32024
UR  - https://www.ams.org/mathscinet-getitem?mr=2038781
UR  - https://doi.org/10.5802/aif.1993
DO  - 10.5802/aif.1993
LA  - en
ID  - AIF_2003__53_6_1819_0
ER  - 
%0 Journal Article
%A González Pérez, Pedro D.
%T Toric embedded resolutions of quasi-ordinary hypersurface singularities
%J Annales de l'Institut Fourier
%D 2003
%P 1819-1881
%V 53
%N 6
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.1993
%R 10.5802/aif.1993
%G en
%F AIF_2003__53_6_1819_0
González Pérez, Pedro D. Toric embedded resolutions of quasi-ordinary hypersurface singularities. Annales de l'Institut Fourier, Volume 53 (2003) no. 6, pp. 1819-1881. doi : 10.5802/aif.1993. http://archive.numdam.org/articles/10.5802/aif.1993/

[A'C-Ok] N. A' Campo; M. Oka Geometry of plane curves via Tschirnhausen resolution tower, Osaka J. Math, Volume 33 (1996), pp. 1003-1033 | MR | Zbl

[A-M] S.S. Abhyankar; T. Moh Newton-Puiseux Expansion and Generalized Tschirnhausen Transformation I-II, J. reine angew. Math, Volume 260 (1973), pp. 47-83 | DOI | MR | Zbl

[A-M] S.S. Abhyankar; T. Moh Newton-Puiseux expansion and generalized Tschirnhausen transformation. I, II., J. Reine Angew. Math., Volume 261 (1973), pp. 29-54 | MR | Zbl

[A1] S.S. Abhyankar On the ramification of algebraic functions., Amer. J. Math., Volume 77 (1955), pp. 575-592 | DOI | MR | Zbl

[A2] S.S. Abhyankar Inversion and invariance of characteristic pairs, Amer. J. Math, Volume 89 (1967), pp. 363-372 | DOI | MR | Zbl

[A3] S.S. Abhyankar Expansion Techniques in Algebraic Geometry, Tata Instit. Fund. Research, Bombay (1977)

[B-M] C. Ban; L. McEwan Canonical resolution of a quasi-ordinary surface singularity, Canad. J. Math., Volume 52 (2000) no. 6, pp. 1149-1163 | DOI | MR | Zbl

[B-P-V] W. Barth; C. Peters; A. Van de Ven Compact Complex Surfaces, Annals of Math. Studies (3), Springer-Verlag, 1984 | MR | Zbl

[Bbk] N. Bourbaki Algebre commutative, Chap. I-IV, Masson, 1981 | MR | Zbl

[Ca] A. Campillo Algebroid Curves in positive characteristic, Lecture Notes in Mathematics, 813, Springer, Berlin, 1980 | MR | Zbl

[Co] D. Cox; H. Hauser, J. Lipman Toric Varieties and Toric Resolutions, Resolution of Singularities. A research textbook in tribute to Oscar Zariski (Progress in Mathematics), Volume 181 (2000), pp. 259-283 | Zbl

[Eg] H. Eggers Polarinvarianten und die Topologie von Kurvensingularitaten, Bonner Mathematische Schriften, Volume 147 (1983) | MR | Zbl

[Ew] G. Ewald Combinatorial Convexity and Algebraic Geometry, Springer-Verlag, 1996 | MR | Zbl

[F] W. Fulton Introduction to Toric Varieties, Annals of Math. Studies, 131, Princeton University Press, 1993 | MR | Zbl

[G-P] J. Gwo\' zdziewicz; A. Ploski On the Approximate Roots of Polynomials, Annales Polonici Mathematici, Volume LX (1995) no. 3, pp. 199-210 | MR | Zbl

[G-T] R. Goldin; B. Teissier; H. Hauser, J. Lipman Resolving singularities of plane analytic branches with one toric morphism, Resolution of Singularities. A research textbook in tribute to Oscar Zariski. (Progress in Mathematics), Volume 181 (2000), pp. 315-340 | Zbl

[Gau] Y-N. Gau Embedded Topological classification of quasi-ordinary singularities, Memoirs of the American Mathematical Society, Volume 388 (1988) | MR | Zbl

[GB-GP] E.R. Garc\'ia; Barroso; P.D. González; Pérez Decomposition in bunches of the critical locus of a quasi-ordinary map (submitted). | Zbl

[GB1] E.R. Garc\'ia; Barroso Invariants des singularités de courbes planes et courbure des fibres de Milnor (1996) Tesis Doctoral, Universidad de La Laguna (Spain)

[GB2] E.R. Garc\'ia; Barroso Sur les courbes polaires d'une courbe plane réduite, Proc. London Math. Soc, Volume 81 (2000) no. 1, pp. 1-28 | DOI | MR | Zbl

[GP-M-N] P.D. González; Pérez; L.J. Mc; Ewan; A. Némethi The zeta function of a quasi-ordinary singularity II (to appear in R. Michler Memorial, Proc. Amer. Math. Soc.) | MR | Zbl

[GP-T] P.D. González; Pérez; B. Teissier Toric embedded resolution of non necessarily normal toric varieties, to appear in C. R. Acad. Sci. Paris, Sér. I Math. | Zbl

[GP1] P.D. González; Pérez Singularités quasi-ordinaires toriques et polyèdre de Newton du discriminant, Canadian J. Math., Volume 52 (2000) no. 2, pp. 348-368 | DOI | MR | Zbl

[GP2] P.D. González; Pérez Quasi-ordinary singularities via toric geometry (2000) (Tesis Doctoral, Universidad de La Laguna)

[GP3] P.D. González; Pérez The semigroup of a quasi-ordinary hypersurface (to appear in J. Inst. Math. Jussieu) | MR

[GS-LJ] G. Gonzalez-Sprinberg; M. Lejeune-Jalabert Modèles canoniques plongés. I, Kodai Math. J., Volume 14 (1991) no. 2, pp. 194-209 | DOI | MR | Zbl

[J] H.W.E. Jung Darstellung der Funktionen eines algebraischen Körpers zweier unabhaängigen Veränderlichen x, y in der Umgebung einer stelle x=a, y=b, J. reine angew. Math., Volume 133 (1908), pp. 289-314 | DOI | JFM

[K-K-M-S] G. Kempf; F. Knudsen; D. Mumford; B. St-Donat Toroidal Embeddings, Springer Lecture Notes in Mathematics, 339, Springer Verlag, 1973 | Zbl

[Kou] A.G. Kouchnirenko Polyèdres de Newton et nombres de Milnor, Inv. Mat, Volume 32 (1976), pp. 1-31 | DOI | MR | Zbl

[L-M-W] D.T. Lê; F. Michel; C. Weber Sur le comportement des polaires associées aux germes de courbes planes, Compositio Math., Volume 72 (1989) no. 1, pp. 87-113 | Numdam | MR | Zbl

[L1] J. Lipman Quasi-ordinary singularities of embedded surfaces (1965) (Thesis, Harvard University)

[L2] J. Lipman Introduction to Resolution of Singularities, Proceedings of Symposia in Pure Mathematics, Volume 29 (1975), pp. 187-230 | MR | Zbl

[L3] J. Lipman Quasi-ordinary singularities of surfaces in 3 , Proceedings of Symposia in Pure Mathematics, Volume 40 (1983) no. 2, pp. 161-172 | MR | Zbl

[L4] J. Lipman Topological invariants of quasi-ordinary singularities, Memoirs of the American Mathematical Society, Volume 388 (1988) | MR | Zbl

[L5] J. Lipman; H. Hauser, J. Lipman Equisingularity and simultaneous resolution of singularities, Resolution of Singularities. A research textbook in tribute to Oscar Zariski. (Progress in Mathematics), Volume 181 (2000), pp. 485-503 | Zbl

[Lau] H. Laufer Normal two dimensional singularities, Annals of Math. Studies, 71, Princenton University Press, 1971 | MR | Zbl

[Le-Ok] D.T. Lê; M. Oka On resolution complexity of plane curves, Kodaira Math. J, Volume 18 (1995), pp. 1-36 | DOI | MR | Zbl

[LJ] M. Lejeune-Jalabert; Lê D\ ung Tráng Sur l'équivalence des singularités des courbes algebro\" \i des planes (coefficients de Newton), Introduction à la théorie des singularités I (1988), pp. 49-154 | Zbl

[LJ-R] M. Lejeune-Jalabert; A. Reguera López Arcs and wedges on sandwiched surface singularities, Amer. J. Math, Volume 121 (1999) no. 6, pp. 1191-1213 | DOI | MR | Zbl

[LJ-R2] M. Lejeune-Jalabert; A. Reguera López Desingularization of both a plane branch C and its monomial curve C Γ (2000) (Manuscript)

[Lu] I. Luengo On the structure of embedded algebroid surfaces, Proceedings of Symposia in Pure Mathematics, Volume 40 (1983), pp. 185-193 | MR | Zbl

[M-N] L.J. McEwan; A. Némethi The zeta function of a quasi-ordinary singularity I (to appear in Compositio Math.) | MR | Zbl

[Me] M. Merle Invariants polaires des courbes planes, Inv. Math., Volume 41 (1977), pp. 103-111 | DOI | MR | Zbl

[Mu] D. Mumford The Red Book on Varieties and Schemes, Lecture Notes in Mathematics, 1358, Springer-Verlag, 1988 | MR | Zbl

[Od] T. Oda Convex Bodies and Algebraic Geometry, Annals of Math. Studies, 131, Springer-Verlag, 1988 | MR | Zbl

[Ok] M. Oka; A. Campillo López and L. Narváez Macarro Geometry of plane curves via toroidal resolution, Algebraic Geometry and Singularities (Progress in Mathematics), Volume 139 (1996) | Zbl

[PP1] P. Popescu-Pampu; F.-V. Kuhlmann, S.Kuhlmann Approximate roots, Valuation Theory and its Applications (Fields Inst. Communications Ser.), Volume vol. II | Zbl

[PP2] P. Popescu-Pampu Arbres de contact des singularités quasi-ordinaires et graphes d'adjacence pour les 3-variétés réelles (2001) (Thèse de Doctorat, Université de Paris 7)

[Re] J.E. Reeve A summary of results on the topological classification of plane algebroid singularities, Rend. Sem. Mat. Univ. e Politec. Torino (1954-55), Volume 14, pp. 159-187 | Zbl

[St] B. Sturmfels Gröbner Bases and Convex Polytopes, University Lecture Series, Vol 8, American Mathematical Society, 1996 | MR | Zbl

[T1] B. Teissier The monomial curve and its deformations. Appendix in [Z6]

[T2] B. Teissier; F.-V. Kuhlmann, S. Kuhlmann Valuations, Deformations and Toric Geometry, Valuation Theory and its Applications. (Fields Inst. Communications Ser.), Volume vol. II | Zbl

[V1] O. Villamayor Constructiveness of Hironaka's resolution., Ann. Sci. Ecole Norm. Sup. (4), Volume 22 (1989) no. 1, pp. 1-32 | Numdam | MR | Zbl

[V2] O. Villamayor On Equiresolution and a question of Zariski, Acta Math, Volume 185 (2000), pp. 123-159 | DOI | MR | Zbl

[W] R.J. Walker Reduction of the Singularities of an Algebraic Surface, Annals of Maths, Volume 36 (1935) no. 2, pp. 336-365 | DOI | JFM | MR

[Wa] C.T.C. Wall Chains on the Eggers tree and polar curves, Revista Mat. Iberoamericana, Volume 19 (2003), pp. 1-10 | MR | Zbl

[Z1] O. Zariski Le probléme de la réduction des singularités d'une variété algébrique, Bull. Sci. Mathématiques, Volume 78 (1954), pp. 31-40 | MR | Zbl

[Z2] O. Zariski The connectedness theorem for birrational transformations, Algebraic Geometry and Topology (Symposium in honor of S. Lefschetz) (1955), pp. 182-188 | Zbl

[Z3] O. Zariski Studies in Equisingularity. I., Amer. J. Math., Volume 87 (1965), pp. 507-536 | MR | Zbl

[Z3] O. Zariski Studies in equisingularity. II., Amer. J. Math., Volume 87 (1965), pp. 972-1006 | MR | Zbl

[Z3] O. Zariski Collected Papers, IV (1979)

[Z4] O. Zariski; Edizioni Cremonese Contributions to the problem of equisingularity, Questions on Algebraic varieties. (C.I.M.E., III ciclo, Varenna 7-17 Settembre 1969) (1970), pp. 261-343 | Zbl

[Z4] O. Zariski Collected papers, IV (1979)

[Z5] O. Zariski Exceptional Singularities of an Algebroid Surface and their Reduction, Atti. Accad. Naz. Lincei Rend., Cl. Sci. Fis. Mat. Natur. (8), Volume 43 (1967), pp. 135-146 | MR | Zbl

[Z5] O. Zariski Collected papers, I (1979)

[Z6] O. Zariski Le problème des modules pour les branches planes, Hermann, Paris, 1986 | MR | Zbl

Cited by Sources: