Tohoku Mathematical Journal

Characterization of 2-dimensional normal Mather-Jacobian log canonical singularities

Kohsuke Shibata and Nguyen Duc Tam

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In this paper we characterize 2-dimensional normal Mather-Jacobian log canonical singularities which are not complete intersections. We prove that a 2-dimensional normal singularity which is not a complete intersection is a Mather-Jacobian log canonical singularity if and only if it is a toric singularity with embedding dimension 4.

Article information

Tohoku Math. J. (2), Volume 71, Number 1 (2019), 123-136.

First available in Project Euclid: 9 March 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J17: Singularities [See also 14B05, 14E15]
Secondary: 14E18: Arcs and motivic integration 14M25: Toric varieties, Newton polyhedra [See also 52B20]

Mather Jacobian singularity surface singularity Toric varieties


Shibata, Kohsuke; Tam, Nguyen Duc. Characterization of 2-dimensional normal Mather-Jacobian log canonical singularities. Tohoku Math. J. (2) 71 (2019), no. 1, 123--136. doi:10.2748/tmj/1552100445.

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  • M. Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88, No. 1, (1966), 129–136.
  • T. de Fernex and R. Docampo, Jacobian discrepancies and rational singularities, arXiv:1106.2172, J. Eur. Math. Soc. (2014), no. 1, 165–199.
  • T. de Fernex and C. Hacon, Singularities on normal varieties, Math. Comp. 145 (2009), 393–414.
  • T. de Fernex, L. Ein and S. Ishii, Divisorial valuations via arcs, Publ. RIMS 44 (2008), 425–448.
  • L. Ein and S. Ishii, Singularities with respect to Mather-Jacobian discrepancies, Commutative Algebra and Noncommutative Algebraic Geometry, Vol. II, 125–168, Math. Sci. Res. Inst. Publ. 68, Cambridge Univ. Press, New York, 2015.
  • L. Ein, S. Ishii and M. Mustaţă, Multiplier ideals via Mather discrepancy, Minimal models and extremal rays (Kyoto, 2011), 9–28, Adv. Stud. Pure Math. 70, Math. Soc. Japan, Tokyo, 2016.
  • L. Ein and M. Mustaţă, Jet schemes and singularities, in Algebraic Geometry–Seattle 2005, Part 2, 505–546, Proc. Sympos. Pure Math. 80, Part 2, Amer. Math. Soc., Providence, RI, 2009.
  • W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol 131, Princeton Univ. Press, Princeton, New Jersey, 1993.
  • S. Ishii, Introduction to Singularities, Springer, 2014.
  • S. Ishii, Mather discrepancy and the arc spaces, arXiv:1106.0345, Ann. Inst. Fourier 63 (2013), 89–111.
  • H. Laufer, Normal two-dimensional singularities, Annals of Mathematics Studies, No. 71, Princeton Univ. Press, Princeton, New Jersey, 1971.
  • O. Riemenschneider, Deformationen von Quotientensingularitäten (nach zyklischen Gruppen), Math. Ann. 209 (1974), 211–248.
  • G. N. Tyurina, Absolute isolatedness of rational singularities and rational triple points, J. Fourier Anal. Appl. 2 (1968), 324–332.
  • J. Voight, Toric surfaces and continued fractions, https://math.dartmouth. edu/~jvoight/notes/cfrac.pdf