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The geometric genus of splice-quotient singularities


Author: Tomohiro Okuma
Journal: Trans. Amer. Math. Soc. 360 (2008), 6643-6659
MSC (2000): Primary 32S25; Secondary 14B05, 14J17
DOI: https://doi.org/10.1090/S0002-9947-08-04559-5
Published electronically: July 22, 2008
MathSciNet review: 2434304
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove a formula for the geometric genus of splice-quotient singularities (in the sense of Neumann and Wahl). This formula enables us to compute the invariant from the resolution graph; in fact, it reduces the computation to that for splice-quotient singularities with smaller resolution graphs. We also discuss the dimension of the first cohomology groups of certain invertible sheaves on a resolution of a splice-quotient singularity.


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  • 1. M. Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129-136. MR 0199191 (33:7340)
  • 2. W. Bruns and J. Herzog, Cohen-Macaulay rings $ ($Revised edition$ )$, Cambridge Stud. Adv. Math., vol. 39, Cambridge Univ. Press, Cambridge, 1998.
  • 3. S. Goto and K.-i. Watanabe, On graded rings. I, J. Math. Soc. Japan 30 (1978), no. 2, 179-213. MR 494707 (81m:13021)
  • 4. H. Laufer, On minimally elliptic singularities, Amer. J. Math. 99 (1977), 1257-1295. MR 0568898 (58:27961)
  • 5. I. Luengo-Velasco, A. Melle-Hernández, and A. Némethi, Links and analytic invariants of superisolated singularities, J. Algebraic Geom. 14 (2005), no. 3, 543-565. MR 2129010 (2005m:32057)
  • 6. A. Némethi, Line bundles associated with normal surface singularities, arXiv:math.AG/0310084.
  • 7. -, ``Weakly'' elliptic Gorenstein singularities of surfaces, Invent. Math. 137 (1999), no. 1, 145-167. MR 1703331 (2000e:32037)
  • 8. A. Némethi and L. I. Nicolaescu, Seiberg-Witten invariants and surface singularities, Geom. Topol. 6 (2002), 269-328. MR 1914570 (2003i:14048)
  • 9. A. Némethi and T. Okuma, On the Casson Invariant Conjecture of Neumann-Wahl, arXiv:math.AG/0610465, to appear in J. Algebraic Geom.
  • 10. W. D. Neumann, Graph $ 3$-manifolds, splice diagrams, singularities, Singularity theory, World Sci. Publ., Hackensack, NJ, 2007, pp. 787-817. MR 2342940
  • 11. -, A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Amer. Math. Soc. 268 (1981), no. 2, 299-344. MR 632532 (84a:32015)
  • 12. W. D. Neumann and J. Wahl, The end curve theorem for normal complex surface singularities, arXiv:0804.4644v1.
  • 13. -, Casson invariant of links of singularities, Comment. Math. Helv. 65 (1990), 58-78. MR 1036128 (91c:57022)
  • 14. -, Universal abelian covers of surface singularities, Trends in singularities, Trends Math., Birkhäuser, Basel, 2002, pp. 181-190. MR 1900786 (2003c:32028)
  • 15. -, Complete intersection singularities of splice type as universal abelian covers, Geom. Topol. 9 (2005), 699-755. MR 2140991 (2006i:32037)
  • 16. -, Complex surface singularities with integral homology sphere links, Geom. Topol. 9 (2005), 757-811. MR 2140992 (2006b:32042)
  • 17. T. Okuma, Universal abelian covers of rational surface singularities, J. London Math. Soc. (2) 70 (2004), 307-324. MR 2078895 (2005e:14006)
  • 18. -, Universal abelian covers of certain surface singularities, Math. Ann. 334 (2006), 753-773. MR 2209255 (2007f:32036)
  • 19. F. Sakai, Weil divisors on normal surfaces, Duke Math. J. 51 (1984), no. 4, 877-887. MR 771385 (86m:14025)
  • 20. R. Stanley, Combinatorics and commutative algebra, second ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1453579 (98h:05001)
  • 21. 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), 681-740. MR 1031224 (91a:14010)

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Additional Information

Tomohiro Okuma
Affiliation: Department of Education, Yamagata University, Yamagata 990-8560, Japan
Email: okuma@e.yamagata-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-08-04559-5
Keywords: Surface singularity, geometric genus, rational homology sphere, splice type singularity, universal abelian cover
Received by editor(s): October 18, 2006
Received by editor(s) in revised form: March 13, 2007
Published electronically: July 22, 2008
Additional Notes: This work was partly supported by the Grant-in-Aid for Young Scientists (B), The Ministry of Education, Culture, Sports, Science and Technology, Japan.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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