Tohoku Mathematical Journal

The maximal ideal cycles over normal surface singularities with ${\Bbb C}^*$-action

Masataka Tomari and Tadashi Tomaru

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


The maximal ideal cycles and the fundamental cycles are defined on the exceptional sets of resolution spaces of normal complex surface singularities. The former (resp. later) is determined by the analytic (resp. topological) structure of the singularities. We study such cycles for normal surface singularities with ${\Bbb C}^*$-action. Assuming the existence of a reduced homogeneous function of the minimal degree, we prove that these two cycles coincide if the coefficients on the central curve of the exceptional set of the minimal good resolution coincide.

Article information

Tohoku Math. J. (2), Volume 69, Number 3 (2017), 415-430.

First available in Project Euclid: 12 September 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32S25: Surface and hypersurface singularities [See also 14J17]
Secondary: 32S10: Invariants of analytic local rings 14D06: Fibrations, degenerations

surface singularities maximal ideal cycles fundamental cycles


Tomari, Masataka; Tomaru, Tadashi. The maximal ideal cycles over normal surface singularities with ${\Bbb C}^*$-action. Tohoku Math. J. (2) 69 (2017), no. 3, 415--430. doi:10.2748/tmj/1505181624.

Export citation


  • M. Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129–136.
  • E. Brieskorn and H. Knörrer, Plane algebraic curves, Birkhäuser-Verlag, Basel, Boston, Stuttgart, 1986.
  • D. J. Dixon, The fundamental divisor of normal double points of surfaces, Pacific J. Math. 80 (1979), no.1, 105–115.
  • A. Fujiki, On isolated singularities with ${\Bbb C}^*$-action, (in Japanese), Master thesis, Kyoto University, 1972.
  • M. Jankins and W. D. Neumann, Lectures on Seifert manifolds, Brandeis Lecture Notes, 2, Brandeis University, Waltham, MA, 1983.
  • U. Karras, On pencils of curves and deformations of minimally elliptic singularities, Math. Ann. 247 (1980), no. 1, 43–65.
  • K. Konno and D. Nagashima, Maximal ideal cycles over normal surface singularities of Brieskorn type, Osaka J. Math. 49 (2012), no. 1, 225–245.
  • H. Laufer, On minimally elliptic singularities, Amer. J. Math. 99 (1977), no. 6, 1257–1295.
  • H. Laufer, On normal two-dimensional double point singularities, Israel J. Math. 31 (1978), no. 3-4, 315–334.
  • H. Laufer, Tangent cones for deformations of two-dimensional quasi-homogeneous singularities, Singularities, (Iowa City, IA, 1986), 183–197, Contemp. Math., 90, Amer. Math. Soc., Providence, RI, 1989.
  • F. N. Meng and T. Okuma, The maximal ideal cycles over Complete intersection surface singularities of Brieskorn type, Kyushu J. Math. 68 (2014), no.1, 121–137.
  • P. Orlik and Ph. Wagreich, Isolated singularities of algebraic surfaces with ${\Bbb C}^*$-action, Ann. of Math. 93 (1971), no. 2, 205–228.
  • H. Pinkham, Normal surface singularities with ${\Bbb C}^*$-action, Math. Ann. 227 (1977), no. 2, 183–193.
  • O. Riemenschneider, Deformationen von Quotientensingularitäten (nach zyklischen Gruppen), Math. Ann. 209 (1974), 211–248.
  • M. Tomari, A $p_g$-formula and elliptic singularities, Publ. Res. Inst. Math. Sci. 21 (1985), no. 2, 297–354.
  • M. Tomari and K.-i. Watanabe, Filtered rings, filtered blowing-ups and normal two-dimensional singularities with “star-shaped” resolution, Publ. Res. Inst. Math. Sci. 25 (1989), no. 5, 681–740.
  • M. Tomari and K.-i. Watanabe, Cyclic covers of normal graded rings, K$\bar o$dai Math. J. 24 (2001), 436–457.
  • T. Tomaru, On Gorenstein surface singularities with fundamental genus $p_f \geqq 2$ which satisfy some minimality conditions, Pacific J. Math. 170 (1995), no. 1, 271–295.
  • T. Tomaru, On Kodaira singularities defined by $z^n=f(x,y)$, Math. Z. 236 (2001), no. 1, 133–149.
  • T. Tomaru, Pinkham-Demazure construction for two dimensional cyclic quotient singularities, Tsukuba J. Math. 25 (2001), no. 1, 75–83.
  • T. Tomaru, Pencil genus for normal surface singularities, J. Math. Soc. Japan 59 (2007), no. 1, 35–80.
  • T. Tomaru, ${\Bbb C}^*$-equivariant degenerations of curves and normal surface singularities with ${\Bbb C}^*$-action, J. Math. Soc. Japan 65 (2013), no. 3, 829–885.
  • Ph. Wagreich, Elliptic singularities of surfaces, Amer. J. Math. 92 (1970), 419–454.
  • S. S.-T. Yau, On maximally elliptic singularities, Trans. Amer. Math. Soc. 257 (1980), no. 2, 269–329.