×

Nonisolated forms of rational triple point singularities of surfaces and their resolutions. (English) Zbl 1368.32019

Let \(S\) be a germ of a normal surface embedded in \({\mathbb C}^N\) with a singularity at the origin, and let \(\pi:\widetilde{S}\to S\) be a resolution of \(S\). The singularity of \(S\) is called rational if \(H^1(\widetilde{S},{\mathcal O}_{\widetilde{S}})=0\). This condition is known to imply a number of combinatorial results on the invariants obtained from the resolution graphs of such surfaces. In the present paper, the authors give a list of non-isolated hypersurface singularities in \({\mathbb C}^3\) such that their normalizations are rational surface singularities of multiplicity \(3\) (those can be defined by three equations in \({\mathbb C}^4\)). Using a method introduced by Oka for isolated complete intersections, they construct the corresponding minimal resolution graphs. They also show that both normal surfaces in \({\mathbb C}^4\) and their non-isolated forms in \({\mathbb C}^3\) are Newton non-degenerate, which means, roughly speaking, that they can be resolved by toric modifications well-behaved with respect to the Newton polygons.

MSC:

32S25 Complex surface and hypersurface singularities
32S45 Modifications; resolution of singularities (complex-analytic aspects)
58K20 Algebraic and analytic properties of mappings on manifolds

Software:

Gfan; SINGULAR
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] F. Aroca, M. Gómez-Morales and K. Shabbir, Torical modification of Newton non-degenerate ideals , preprint, arXiv: · Zbl 1262.14041
[2] M. Artin, On isolated rational singularities of surfaces , Amer. J. Math. 88 (1966), 129-136. · Zbl 0142.18602
[3] W. Barth, C. Peter and A. Van de Ven, Compact complex surfaces , Springer-Verlag, New York, 1984. · Zbl 0718.14023
[4] R. Bondil and D.T. Lê, Résolution des singularités de surfaces par éclatements normalisés \((\)multiplicté, multiplicité polaire, et singularités minimales\()\) , Trends Math., Birkhauser, Basel, 2002.
[5] Z. Chen, R. Du, S.-L. Tan and F. Yu, Cubic equations of rational triple points of dimension two , in American Mathematical Society, Providence, RI, 2007, 63-76. · Zbl 1127.14003
[6] T. de Jong and D. van Straten, A deformation theory for nonisolated singularities , Abh. Math. Sem. Univ. Hamburg 60 (1990), 177-208. · Zbl 0721.32014
[7] W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann, Singular 3-1-3- A computer algebra system for polynomial computations , Kaiserslautern, Germany, 2011, available at http://www.singular.uni-kl.de.
[8] G. Ewald, Combinatorial convexity and algebraic geometry , Grad. Texts Math. 168 , Springer Verlag, Berlin, 1996.
[9] G. Ficher, Complex analytic geometry , Lect. Notes Math. 538 , Springer-Verlag, Berlin, 1976.
[10] W. Fulton, Introduction to toric varieties , Princeton University Press, Princeton, 1993. · Zbl 0813.14039
[11] H. Grauert, Uber Modifikationen und exzeptionelle analytische Mengen , Math. Ann. 146 (1962), 331-368. · Zbl 0173.33004
[12] R. Hartshorne, Algebraic geometry , Grad. Texts Math. 52 , Springer, New York, 1977. · Zbl 0367.14001
[13] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero I-II, Ann. Math. 79 (1964), 109-203: ibid. 79 (1964), 205-326. · Zbl 0122.38603
[14] A.N. Jensen, Gfan, A software system for Gröbner fans and tropical varieties , 2009, Aarhus, Denmark, available at http://home.imf.au.dk/ jensen/software/gfan/gfan.html.
[15] H.B. Laufer, Normal two-dimensional singularities , Ann. Math. Stud. 71 , Princeton University Press, Princeton, N.J., 1971. · Zbl 0245.32005
[16] —-, On rational singularities , Amer. J. Math. 94 (1972), 597-608. · Zbl 0251.32002
[17] —-, Taut two-dimensional singularities , Math. Ann. 205 (1973), 131-164. · Zbl 0281.32010
[18] D.T. Lê and B. Teissier, Variêtês polaires locales et classes de Chern des variêtês singuliêres , Ann. Math. 114 (1981), 457-491. · Zbl 0488.32004
[19] D.T. Lê and M. Tosun, Combinatorics of rational singularities , Comm. Math. Helv. 79 (2004), 582-604. · Zbl 1060.32016
[20] R. Miranda, Triple covers in algebraic geometry , Amer. J. Math. 107 (1985), 1123-1158. · Zbl 0611.14011
[21] D. Mond, Some remarks on the geometry and classification of germs of maps from surfaces to \(3\)-spaces , Topology 26 (1987), 361-383. · Zbl 0654.32008
[22] D. Mumford, Algebaric geometry I: Complex projective varieties , Springer Verlag, New York, 1976. · Zbl 0356.14002
[23] Z. Oer, A. Özkan and M. Tosun, On the classification of rational singularities of surfaces , Int. J. Pure Appl. Math. 41 (2007), 85-110. · Zbl 1137.14007
[24] M. Oka, Non-degenerate complete intersection singularity , in Actualites mathematiques , Hermann, Paris, 1997. · Zbl 0930.14034
[25] J. Stevens, Partial resolutions of rational quadruple points , Inter. J. Math. 2 (1991), 205-221. · Zbl 0726.14026
[26] —-, On the classification of rational surface singularities , preprint, arXiv:
[27] S.-L. Tan, Triple covers on smooth algebraic varieties , in Geometry and nonlinear partial differential equations , American Mathematical Society and International Press, 2002.
[28] G.N. Tyurina, Absolute isolatedness of rational singularities and rational triple points , Fonc. Anal. Appl. 2 (1968), 324-332. · Zbl 0176.50804
[29] M. Tosun, Tyurina components and rational cycles for rational singularities , Turkish J. Math. 23 (1999), 361-374. · Zbl 0970.32013
[30] A.N. Varchenko, Zeta-function of monodromy and Newton’s diagram , Invent. Math. 37 (1976), 253-262. · Zbl 0333.14007
[31] J.M. Wahl, Equations defining rational singularities , Ann. Sci. Ecol. Norm. 10 (1977), 231-264. · Zbl 0367.14004
[32] O. Zariski, Polynomial ideals defined by infinitely near base points , Amer. J. Math. 60 (1938), 151-204. · Zbl 0018.20101
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.