Abstract
In a previous work we have introduced and studied a special kind of toric resolution, the so-called embedded Q-resolution, which essentially consists in allowing the final ambient space to contain abelian quotient singularities. Here we explicitly compute an embedded Q-resolution of a Yomdin-Lê surface singularity (V, 0) in terms of a (global) embedded Q-resolution of its tangent cone by means of just weighted blow-ups at points. The generalized A’Campo’s formula in this setting is applied so as to compute the characteristic polynomial. As a consequence, an exceptional divisor in the resolution of (V, 0), apart from the first one which might be special, contributes to its complex monodromy if and only if so does the corresponding divisor in the tangent cone. Thus the resolution obtained is optimal in the sense that the weights can be chosen so that every exceptional divisor in the Q-resolution of (V, 0), except perhaps the first one, contributes to its monodromy.
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References
E. Artal-Bartolo, Combinatorics and topology of line arrangements in the complex projective plane, Proceedings of the American Mathematical Society 121 (1994), 385–390.
E. Artal-Bartolo, Forme de Jordan de la monodromie des singularités superisolées de surfaces, Memoirs of the American Mathematical Society 109 (1994).
E. Artal-Bartolo, I. Luengo and A. Melle-Hernández, Superisolated surface singularities, in Singularities and Computer Algebra, London Mathematical Society Lecture Note Series, Vol. 324, Cambridge University Press, Cambridge, 2006, pp. 13–39.
E. Artal-Bartolo, J. Martín-Morales and J. Ortigas-Galindo, Cartier and Weil divisors on varieties with quotient singularities, ArXiv e-prints (2011).
E. Artal-Bartolo, J. Martín-Morales and J. Ortigas-Galindo, Intersection theory on abelian-quotient V -surfaces and Q-resolutions, ArXiv e-prints (2011).
H. Esnault, Fibre de Milnor d’un cône sur une courbe plane singulière, Inventiones Mathematicae 68 (1982), 477–496.
S. M. Gusein-Zade, I. Luengo and A. Melle-Hernández, Partial resolutions and the zetafunction of a singularity, Commentarii Mathematici Helvetici 72 (1997), 244–256.
D. T. Lê, Ensembles analytiques complexes avec lieu singulier de dimension un (d’après I. N. Iomdine), in Seminar on Singularities (Paris, 1976/1977), Publications Mathématiques de l’Université Paris VII, Vol. 7, Université Paris VII, Paris, 1980, pp. 87–95.
I. Luengo, The µ-constant stratum is not smooth, Inventiones Mathematicae 90 (1987), 139–152.
J. Martín-Morales, Embedded Q-Resolutions and Yomdin-Lê Surface Singularities, Ph.D. dissertation, IUMA-University of Zaragoza, December 2011, URL: http://cud.unizar.es/martin.
J. Martín-Morales, Monodromy zeta function formula for embedded Q -resolutions, Revista Matemática Iberoamerican 29 (2013), 939–967.
J. Martín-Morales, Semistable reduction of a normal crossing ℚ-divisor, available at http://arxiv.org/abs/1401.3926
A. Melle-Hernández, Milnor numbers for surface singularities, Israel Journal of Mathematics 115 (2000), 29–50.
D. Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Institut de Hautes Études Scientifiqus. Publications Mathématiques 9 (1961), 5–22.
D. Siersma, The monodromy of a series of hypersurface singularities, Commentarii Mathematici Helvetici 65 (1990), 181–197.
J. H. M. Steenbrink, Mixed Hodge structure on the vanishing cohomology, in Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 525–563.
J. Stevens, On the µ-constant stratum and the V-filtration: an example, Mathematische Zeitschrift 201 (1989), 139–144.
Y. Yomdin, Complex surfaces with a one-dimensional set of singularities, Sibirskiĭ Matematicheskiĭ Žurnal 15 (1974), 1061–1082, 1181.
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The author is partially supported by the Spanish Ministry of Education MTM2010-21740-C02-02, E15 Grupo Consolidado Geometría from the Gobierno de Aragón, FQM-333 from Junta de Andalucía, and PRI-AIBDE-2011-0986 Acción Integrada hispano-alemana.
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Martín-Morales, J. Embedded Q-resolutions for Yomdin-Lê surface singularities. Isr. J. Math. 204, 97–143 (2014). https://doi.org/10.1007/s11856-014-1078-z
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DOI: https://doi.org/10.1007/s11856-014-1078-z