×

Toric embedded resolutions of quasi-ordinary singularities. (English) Zbl 1052.32024

A germ of a complex analytic variety is quasi-ordinary if there exists a finite projection to the complex affine space with discriminant locus contained in a normal crossing divisor. Some properties of complex analytic curve singularities generalize to quasi-ordinary singularities in higher dimensions, for example the existence of fractional power series parametrization, as well as the existence of some distinguished, {characteristic} monomials in the parametrization.
The paper gives two different affirmative solutions to a problem of Lipman: do the characteristic monomials of a reduced hypersurface quasi-ordinary singularity determine a procedure of embedded resolution of the singularity?
The first procedure builds a sequence of toric morphisms depending only on the characteristic monomials. Along the way characteristic monomials are defined for toric quasi-ordinary hypersurface singularities, and their properties are studied. The second procedure generalizes a method of Goldin and Tessier for plane branches. A key step is the re-embedding of the germ in a larger affine space using certain approximate roots of a Weierstrass polynomial. In the last two sections the two procedures are compared and a detailed example is worked out.

MSC:

32S25 Complex surface and hypersurface singularities
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
32S45 Modifications; resolution of singularities (complex-analytic aspects)

Citations:

Zbl 0970.14011
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] Geometry of plane curves via tschirnhausen resolution tower, Osaka J. Math, 33, 1003-1033, (1996) · Zbl 0904.14014
[2] Newton-Puiseux expansion and generalized tschirnhausen transformation I-II, J. reine angew. Math, 260, 47-83, (1973) · Zbl 0272.12102
[3] Newton-Puiseux expansion and generalized tschirnhausen transformation. I, II., J. Reine Angew. Math., 261, 29-54, (1973) · Zbl 0272.12102
[4] On the ramification of algebraic functions., Amer. J. Math., 77, 575-592, (1955) · Zbl 0064.27501
[5] Inversion and invariance of characteristic pairs, Amer. J. Math, 89, 363-372, (1967) · Zbl 0162.34103
[6] Expansion techniques in algebraic geometry, Tata Instit. Fund. Research, Bombay, (1977)
[7] Canonical resolution of a quasi-ordinary surface singularity, Canad. J. Math., 52, 6, 1149-1163, (2000) · Zbl 1002.14003
[8] Compact Complex Surfaces, (1984), Springer-Verlag · Zbl 0718.14023
[9] Algebre commutative, Chap. I-IV, (1981), Masson · Zbl 0498.12001
[10] Algebroid Curves in positive characteristic, 813, (1980), Springer, Berlin · Zbl 0451.14010
[11] Toric varieties and toric resolutions, Resolution of Singularities. A research textbook in tribute to Oscar Zariski, 181, 259-283, (2000), Birkhäuser-Verlag · Zbl 0969.14035
[12] Polarinvarianten und die topologie von kurvensingularitaten, Bonner Mathematische Schriften, 147, (1983) · Zbl 0559.14018
[13] Combinatorial Convexity and Algebraic Geometry, (1996), Springer-Verlag · Zbl 0869.52001
[14] Introduction to Toric Varieties, 131, (1993), Princeton University Press · Zbl 0813.14039
[15] On the approximate roots of polynomials, Annales Polonici Mathematici, LX, 3, 199-210, (1995) · Zbl 0826.13012
[16] Resolving singularities of plane analytic branches with one toric morphism, Resolution of Singularities. A research textbook in tribute to Oscar Zariski., 181, 315-340, (2000), Birkhäuser-Verlag · Zbl 0995.14002
[17] Embedded topological classification of quasi-ordinary singularities, Memoirs of the American Mathematical Society, 388, (1988) · Zbl 0658.14004
[18] Decomposition in bunches of the critical locus of a quasi-ordinary map (submitted). · Zbl 1079.14059
[19] Invariants des singularités de courbes planes et courbure des fibres de Milnor, (1996)
[20] Sur LES courbes polaires d’une courbe plane réduite, Proc. London Math. Soc, 81, 1, 1-28, (2000) · Zbl 1041.14008
[21] The zeta function of a quasi-ordinary singularity II · Zbl 1080.14002
[22] Toric embedded resolution of non necessarily normal toric varieties, to appear in C. R. Acad. Sci. Paris, Sér. I Math. · Zbl 1052.14062
[23] Singularités quasi-ordinaires toriques et polyèdre de Newton du discriminant, Canadian J. Math., 52, 2, 348-368, (2000) · Zbl 0970.14027
[24] Quasi-ordinary singularities via toric geometry, (2000)
[25] The semigroup of a quasi-ordinary hypersurface · Zbl 1036.32020
[26] Modèles canoniques plongés. I, Kodai Math. J., 14, 2, 194-209, (1991) · Zbl 0772.14008
[27] Darstellung der funktionen eines algebraischen Körpers zweier unabhaängigen veränderlichen \(x, y\) in der umgebung einer stelle \(x=a, y=b,\) J. reine angew. Math., 133, 289-314, (1908) · JFM 39.0493.01
[28] Toroidal Embeddings, 339, (1973), Springer Verlag · Zbl 0271.14017
[29] Polyèdres de Newton et nombres de Milnor, Inv. Mat, 32, 1-31, (1976) · Zbl 0328.32007
[30] Sur le comportement des polaires associées aux germes de courbes planes, Compositio Math., 72, 1, 87-113, (1989) · Zbl 0705.32021
[31] Quasi-ordinary singularities of embedded surfaces, (1965)
[32] Introduction to resolution of singularities, Proceedings of Symposia in Pure Mathematics, 29, 187-230, (1975) · Zbl 0306.14007
[33] Quasi-ordinary singularities of surfaces in \(\mathbb{C}^3,\) Proceedings of Symposia in Pure Mathematics, 40, 2, 161-172, (1983) · Zbl 0521.14014
[34] Topological invariants of quasi-ordinary singularities, Memoirs of the American Mathematical Society, 388, (1988) · Zbl 0658.14003
[35] Equisingularity and simultaneous resolution of singularities, Resolution of Singularities. A research textbook in tribute to Oscar Zariski., 181, 485-503, (2000), Birkhäuser-Verlag · Zbl 0970.14011
[36] Normal two dimensional singularities, 71, (1971), Princenton University Press · Zbl 0245.32005
[37] On resolution complexity of plane curves, Kodaira Math. J, 18, 1-36, (1995) · Zbl 0844.14010
[38] Sur l’équivalence des singularités des courbes algebro\" \i des planes (coefficients de Newton), Introduction à la théorie des singularités I, 49-154, (1988), Hermann, Paris · Zbl 0699.14036
[39] Arcs and wedges on sandwiched surface singularities, Amer. J. Math, 121, 6, 1191-1213, (1999) · Zbl 0960.14015
[40] Desingularization of both a plane branch \(C\) and its monomial curve \(C^Γ, (2000)\)
[41] On the structure of embedded algebroid surfaces, Proceedings of Symposia in Pure Mathematics, 40, 185-193, (1983) · Zbl 0527.14032
[42] The zeta function of a quasi-ordinary singularity I · Zbl 1066.14004
[43] Invariants polaires des courbes planes, Inv. Math., 41, 103-111, (1977) · Zbl 0371.14003
[44] The Red Book on Varieties and Schemes, 1358, (1988), Springer-Verlag · Zbl 0658.14001
[45] Convex Bodies and Algebraic Geometry, 131, (1988), Springer-Verlag · Zbl 0628.52002
[46] Geometry of plane curves via toroidal resolution, Algebraic Geometry and Singularities, 139, (1996), Birkhäuser, Basel · Zbl 0857.14014
[47] Approximate roots, Valuation Theory and its Applications, vol. II · Zbl 1036.13017
[48] Arbres de contact des singularités quasi-ordinaires et graphes d’adjacence pour les 3-variétés réelles, (2001)
[49] A summary of results on the topological classification of plane algebroid singularities, Rend. Sem. Mat. Univ. e Politec. Torino (1954-55), 14, 159-187 · Zbl 0067.12904
[50] Gröbner Bases and Convex Polytopes, Vol 8, (1996), American Mathematical Society · Zbl 0856.13020
[51] The monomial curve and its deformations. Appendix in [Z6]
[52] Valuations, deformations and toric geometry, Valuation Theory and its Applications., vol. II · Zbl 1061.14016
[53] Constructiveness of Hironaka’s resolution., Ann. Sci. Ecole Norm. Sup. (4), 22, 1, 1-32, (1989) · Zbl 0675.14003
[54] On equiresolution and a question of Zariski, Acta Math, 185, 123-159, (2000) · Zbl 0989.32004
[55] Reduction of the singularities of an algebraic surface, Annals of Maths, 36, 2, 336-365, (1935) · JFM 61.0705.02
[56] Chains on the eggers tree and polar curves, Revista Mat. Iberoamericana, 19, 1-10, (2003) · Zbl 1057.14032
[57] Le probléme de la réduction des singularités d’une variété algébrique, Bull. Sci. Mathématiques, 78, 31-40, (1954) · Zbl 0055.38802
[58] The connectedness theorem for birrational transformations, Algebraic Geometry and Topology (Symposium in honor of S. Lefschetz), 182-188, (1955), Princenton University Press · Zbl 0087.35601
[59] Studies in equisingularity. I., Amer. J. Math., 87, 507-536, (1965) · Zbl 0132.41601
[60] Studies in equisingularity. II., Amer. J. Math., 87, 972-1006, (1965) · Zbl 0146.42502
[61] Collected Papers, IV, (1979)
[62] Contributions to the problem of equisingularity, Questions on Algebraic varieties. (C.I.M.E., III ciclo, Varenna 7-17 Settembre 1969), 261-343, (1970), Roma · Zbl 0204.54503
[63] Collected papers, IV, (1979)
[64] Exceptional singularities of an algebroid surface and their reduction, Atti. Accad. Naz. Lincei Rend., Cl. Sci. Fis. Mat. Natur. (8), 43, 135-146, (1967) · Zbl 0168.18903
[65] Collected papers, I, (1979)
[66] Le problème des modules pour les branches planes, (1986), Hermann, Paris · Zbl 0592.14010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.