Rocky Mountain Journal of Mathematics

Nonisolated forms of rational triple point singularities of surfaces and their resolutions

A. Altıntaş Sharland, G. Çevik, and M. Tosun

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We give a list of nonisolated hypersurface singularities of which normalizations are the rational triple point singularities of surfaces and construct their minimal resolution graphs by a method introduced by Oka for isolated complete intersection singularities. We also show that nonisolated forms of rational triple point singularities and their normalizations are both Newton non-degenerate.

Article information

Rocky Mountain J. Math., Volume 46, Number 2 (2016), 357-388.

First available in Project Euclid: 26 July 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58K20: Algebraic and analytic properties of mappings

Rational singularities nonisolated hypersurfaces Newton polygon resolution graphs


Sharland, A. Altıntaş; Çevik, G.; Tosun, M. Nonisolated forms of rational triple point singularities of surfaces and their resolutions. Rocky Mountain J. Math. 46 (2016), no. 2, 357--388. doi:10.1216/RMJ-2016-46-2-357.

Export citation


  • F. Aroca, M. Gómez-Morales and K. Shabbir, Torical modification of Newton non-degenerate ideals, preprint, arXiv:1209.5104.
  • M. Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129–136.
  • W. Barth, C. Peter and A. Van de Ven, Compact complex surfaces, Springer-Verlag, New York, 1984.
  • 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.
  • 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.
  • T. de Jong and D. van Straten, A deformation theory for nonisolated singularities, Abh. Math. Sem. Univ. Hamburg 60 (1990), 177–208.
  • 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
  • G. Ewald, Combinatorial convexity and algebraic geometry, Grad. Texts Math. 168, Springer Verlag, Berlin, 1996.
  • G. Ficher, Complex analytic geometry, Lect. Notes Math. 538, Springer-Verlag, Berlin, 1976.
  • W. Fulton, Introduction to toric varieties, Princeton University Press, Princeton, 1993.
  • H. Grauert, Uber Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331–368.
  • R. Hartshorne, Algebraic geometry, Grad. Texts Math. 52, Springer, New York, 1977.
  • 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.
  • A.N. Jensen, Gfan, A software system for Gröbner fans and tropical varieties, 2009, Aarhus, Denmark, available at jensen/software/gfan/gfan.html.
  • H.B. Laufer, Normal two-dimensional singularities, Ann. Math. Stud. 71, Princeton University Press, Princeton, N.J., 1971.
  • ––––, On rational singularities, Amer. J. Math. 94 (1972), 597–608.
  • ––––, Taut two-dimensional singularities, Math. Ann. 205 (1973), 131–164.
  • D.T. Lê and B. Teissier, Variêtês polaires locales et classes de Chern des variêtês singuliêres, Ann. Math. 114 (1981), 457–491.
  • D.T. Lê and M. Tosun, Combinatorics of rational singularities, Comm. Math. Helv. 79 (2004), 582–604.
  • R. Miranda, Triple covers in algebraic geometry, Amer. J. Math. 107 (1985), 1123–1158.
  • D. Mond, Some remarks on the geometry and classification of germs of maps from surfaces to $3$-spaces, Topology 26 (1987), 361–383.
  • D. Mumford, Algebaric geometry I: Complex projective varieties, Springer Verlag, New York, 1976.
  • Z. Oer, A. Özkan and M. Tosun, On the classification of rational singularities of surfaces, Int. J. Pure Appl. Math. 41 (2007), 85–110.
  • M. Oka, Non-degenerate complete intersection singularity, in Actualites mathematiques, Hermann, Paris, 1997.
  • J. Stevens, Partial resolutions of rational quadruple points, Inter. J. Math. 2 (1991), 205–221.
  • ––––, On the classification of rational surface singularities, preprint, arXiv:1204.0269.
  • S.-L. Tan, Triple covers on smooth algebraic varieties, in Geometry and nonlinear partial differential equations, American Mathematical Society and International Press, 2002.
  • G.N. Tyurina, Absolute isolatedness of rational singularities and rational triple points, Fonc. Anal. Appl. 2 (1968), 324–332.
  • M. Tosun, Tyurina components and rational cycles for rational singularities, Turkish J. Math. 23 (1999), 361–374.
  • A.N. Varchenko, Zeta-function of monodromy and Newton's diagram, Invent. Math. 37 (1976), 253–262.
  • J.M. Wahl, Equations defining rational singularities, Ann. Sci. Ecol. Norm. 10 (1977), 231–264.
  • O. Zariski, Polynomial ideals defined by infinitely near base points, Amer. J. Math. 60 (1938), 151–204.