Toric embedded resolutions of quasi-ordinary hypersurface singularities

González Pérez, Pedro D.

González Pérez, Pedro D.

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

Abstract
Résumé

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.

MR 2038781 | Zbl 1052.32024 | 1 citation in Numdam

DOI : https://doi.org/10.5802/aif.1993
Classification: 32S15, 32S45, 14M25, 14E15
Keywords: singularities, embedded resolution, discriminant, topological type

@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {53},
     number = {6},
     year = {2003},
     pages = {1819-1881},
     doi = {10.5802/aif.1993},
     zbl = {1052.32024},
     mrnumber = {2038781},
     language = {en},
     url = {http://www.numdam.org/item/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://www.numdam.org/item/AIF_2003__53_6_1819_0/

References

[A'C-Ok] N. A' Campo; M. Oka Geometry of plane curves via Tschirnhausen resolution tower, Osaka J. Math, Tome 33 (1996), pp. 1003-1033 | MR 1435467 | Zbl 0904.14014

[A-M] S.S. Abhyankar; T. Moh Newton-Puiseux Expansion and Generalized Tschirnhausen Transformation I-II, J. reine angew. Math, Tome 260 (1973), pp. 47-83 | Article | MR 337955 | Zbl 0272.12102

[A-M] S.S. Abhyankar; T. Moh Newton-Puiseux expansion and generalized Tschirnhausen transformation. I, II., J. Reine Angew. Math., Tome 261 (1973), pp. 29-54 | MR 337955 | Zbl 0272.12102

[A1] S.S. Abhyankar On the ramification of algebraic functions., Amer. J. Math., Tome 77 (1955), pp. 575-592 | Article | MR 71851 | Zbl 0064.27501

[A2] S.S. Abhyankar Inversion and invariance of characteristic pairs, Amer. J. Math, Tome 89 (1967), pp. 363-372 | Article | MR 220732 | Zbl 0162.34103

[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., Tome 52 (2000) no. 6, pp. 1149-1163 | Article | MR 1794300 | Zbl 1002.14003

[B-P-V] W. Barth; C. Peters; A. Van De Ven Compact Complex Surfaces, Springer-Verlag, Annals of Math. Studies (3) (1984) | MR 749574 | Zbl 0718.14023

[Bbk] N. Bourbaki Algebre commutative, Masson Tome Chap. I-IV (1981) | MR 643362 | Zbl 0498.12001

[Ca] A. Campillo Algebroid Curves in positive characteristic, Springer, Berlin, Lecture Notes in Mathematics, Tome 813 (1980) | MR 584440 | Zbl 0451.14010

[Co] D. Cox; H. Hauser, J. Lipman, F.Oort And A. Quiros. Toric Varieties and Toric Resolutions, Resolution of Singularities. A research textbook in tribute to Oscar Zariski, Birkhäuser-Verlag (Progress in Mathematics) Tome 181 (2000), pp. 259-283 | Zbl 0969.14035

[Eg] H. Eggers Polarinvarianten und die Topologie von Kurvensingularitaten, Bonner Mathematische Schriften, Tome 147 (1983) | MR 701391 | Zbl 0559.14018

[Ew] G. Ewald Combinatorial Convexity and Algebraic Geometry, Springer-Verlag (1996) | MR 1418400 | Zbl 0869.52001

[F] W. Fulton Introduction to Toric Varieties, Princeton University Press, Annals of Math. Studies, Tome 131 (1993) | MR 1234037 | Zbl 0813.14039

[G-P] J. Gwo\' Zdziewicz; A. Ploski On the Approximate Roots of Polynomials, Annales Polonici Mathematici, Tome LX (1995) no. 3, pp. 199-210 | MR 1316488 | Zbl 0826.13012

[G-T] R. Goldin; B. Teissier; H. Hauser, J. Lipman, F.Oort And A. Quiros. Resolving singularities of plane analytic branches with one toric morphism, Resolution of Singularities. A research textbook in tribute to Oscar Zariski., Birkhäuser-Verlag (Progress in Mathematics) Tome 181 (2000), pp. 315-340 | Zbl 0995.14002

[Gau] Y-N. Gau Embedded Topological classification of quasi-ordinary singularities, Memoirs of the American Mathematical Society, Tome 388 (1988) | MR 954948 | Zbl 0658.14004

[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 1079.14059

[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, Tome 81 (2000) no. 1, pp. 1-28 | Article | MR 1756330 | Zbl 01696272

[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 1986117 | Zbl 1080.14002

[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 1052.14062

[GP1] P.D. González; Pérez Singularités quasi-ordinaires toriques et polyèdre de Newton du discriminant, Canadian J. Math., Tome 52 (2000) no. 2, pp. 348-368 | Article | MR 1755782 | Zbl 0970.14027

[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 1990220

[GS-LJ] G. Gonzalez-Sprinberg; M. Lejeune-Jalabert Modèles canoniques plongés. I, Kodai Math. J., Tome 14 (1991) no. 2, pp. 194-209 | Article | MR 1123416 | Zbl 0772.14008

[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., Tome 133 (1908), pp. 289-314 | Article | JFM 39.0493.01

[K-K-M-S] G. Kempf; F. Knudsen; D. Mumford; B. St-Donat Toroidal Embeddings, Springer Verlag, Springer Lecture Notes in Mathematics, Tome 339 (1973) | Zbl 0271.14017

[Kou] A.G. Kouchnirenko Polyèdres de Newton et nombres de Milnor, Inv. Mat, Tome 32 (1976), pp. 1-31 | Article | MR 419433 | Zbl 0328.32007

[L-M-W] D.T. Lê; F. Michel; C. Weber Sur le comportement des polaires associées aux germes de courbes planes, Compositio Math., Tome 72 (1989) no. 1, pp. 87-113 | Numdam | MR 1026330 | Zbl 0705.32021

[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, Tome 29 (1975), pp. 187-230 | MR 389901 | Zbl 0306.14007

[L3] J. Lipman Quasi-ordinary singularities of surfaces in $ℂ^{3}$ , Proceedings of Symposia in Pure Mathematics, Tome 40 (1983) no. 2, pp. 161-172 | MR 713245 | Zbl 0521.14014

[L4] J. Lipman Topological invariants of quasi-ordinary singularities, Memoirs of the American Mathematical Society, Tome 388 (1988) | MR 954947 | Zbl 0658.14003

[L5] J. Lipman; H. Hauser, J. Lipman, F.Oort And A. Quiros. Equisingularity and simultaneous resolution of singularities, Resolution of Singularities. A research textbook in tribute to Oscar Zariski., Birkhäuser-Verlag (Progress in Mathematics) Tome 181 (2000), pp. 485-503 | Zbl 0970.14011

[Lau] H. Laufer Normal two dimensional singularities, Princenton University Press, Annals of Math. Studies, Tome 71 (1971) | MR 320365 | Zbl 0245.32005

[Le-Ok] D.T. Lê; M. Oka On resolution complexity of plane curves, Kodaira Math. J, Tome 18 (1995), pp. 1-36 | Article | MR 1317003 | Zbl 0844.14010

[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, Hermann, Paris (1988), pp. 49-154 | Zbl 0699.14036

[LJ-R] M. Lejeune-Jalabert; A. Reguera López Arcs and wedges on sandwiched surface singularities, Amer. J. Math, Tome 121 (1999) no. 6, pp. 1191-1213 | Article | MR 1719822 | Zbl 0960.14015

[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, Tome 40 (1983), pp. 185-193 | MR 713247 | Zbl 0527.14032

[M-N] L.J. Mcewan; A. Némethi The zeta function of a quasi-ordinary singularity I (to appear in Compositio Math.) | MR 1986117 | Zbl 1066.14004

[Me] M. Merle Invariants polaires des courbes planes, Inv. Math., Tome 41 (1977), pp. 103-111 | Article | MR 460336 | Zbl 0371.14003

[Mu] D. Mumford The Red Book on Varieties and Schemes, Springer-Verlag, Lecture Notes in Mathematics, Tome 1358 (1988) | MR 971985 | Zbl 0658.14001

[Od] T. Oda Convex Bodies and Algebraic Geometry, Springer-Verlag, Annals of Math. Studies, Tome 131 (1988) | MR 922894 | Zbl 0628.52002

[Ok] M. Oka; A. Campillo López And L. Narváez Macarro Geometry of plane curves via toroidal resolution, Algebraic Geometry and Singularities, Birkhäuser, Basel (Progress in Mathematics) Tome 139 (1996) | Zbl 0857.14014

[PP1] P. Popescu-Pampu; F.-V. Kuhlmann, S.Kuhlmann, M. Marshall Eds Approximate roots, Valuation Theory and its Applications (Fields Inst. Communications Ser.) Tome vol. II | Zbl 1036.13017

[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), Tome 14, pp. 159-187 | Zbl 0067.12904

[St] B. Sturmfels Gröbner Bases and Convex Polytopes, American Mathematical Society, University Lecture Series, Tome Vol 8 (1996) | MR 1363949 | Zbl 0856.13020

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

[T2] B. Teissier; F.-V. Kuhlmann, S. Kuhlmann, M. Marshall Eds. Valuations, Deformations and Toric Geometry, Valuation Theory and its Applications. (Fields Inst. Communications Ser.) Tome vol. II | Zbl 1061.14016

[V1] O. Villamayor Constructiveness of Hironaka's resolution., Ann. Sci. Ecole Norm. Sup. (4), Tome 22 (1989) no. 1, pp. 1-32 | Numdam | MR 985852 | Zbl 0675.14003

[V2] O. Villamayor On Equiresolution and a question of Zariski, Acta Math, Tome 185 (2000), pp. 123-159 | Article | MR 1794188 | Zbl 0989.32004

[W] R.J. Walker Reduction of the Singularities of an Algebraic Surface, Annals of Maths, Tome 36 (1935) no. 2, pp. 336-365 | Article | JFM 61.0705.02 | MR 1503227

[Wa] C.T.C. Wall Chains on the Eggers tree and polar curves, Revista Mat. Iberoamericana, Tome 19 (2003), pp. 1-10 | MR 2023205 | Zbl 1057.14032

[Z1] O. Zariski Le probléme de la réduction des singularités d'une variété algébrique, Bull. Sci. Mathématiques, Tome 78 (1954), pp. 31-40 | MR 62474 | Zbl 0055.38802

[Z2] O. Zariski The connectedness theorem for birrational transformations, Algebraic Geometry and Topology (Symposium in honor of S. Lefschetz), Princenton University Press (1955), pp. 182-188 | Zbl 0087.35601

[Z3] O. Zariski Studies in Equisingularity. I., Amer. J. Math., Tome 87 (1965), pp. 507-536 | MR 177985 | Zbl 0132.41601

[Z3] O. Zariski Studies in equisingularity. II., Amer. J. Math., Tome 87 (1965), pp. 972-1006 | MR 191898 | Zbl 0146.42502

[Z3] O. Zariski Collected Papers Tome 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), Roma (1970), pp. 261-343 | Zbl 0204.54503

[Z4] O. Zariski Collected papers Tome 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), Tome 43 (1967), pp. 135-146 | MR 229648 | Zbl 0168.18903

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

[Z6] O. Zariski Le problème des modules pour les branches planes, Hermann, Paris (1986) | MR 861277 | Zbl 0592.14010