## The geometric genus of splice-quotient singularities

HTML articles powered by AMS MathViewer

- by Tomohiro Okuma PDF
- Trans. Amer. Math. Soc.
**360**(2008), 6643-6659 Request permission

## 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.## References

- Michael Artin,
*On isolated rational singularities of surfaces*, Amer. J. Math.**88**(1966), 129–136. MR**199191**, DOI 10.2307/2373050 - W. Bruns and J. Herzog,
*Cohen-Macaulay rings $($Revised edition$)$*, Cambridge Stud. Adv. Math., vol. 39, Cambridge Univ. Press, Cambridge, 1998. - Shiro Goto and Keiichi Watanabe,
*On graded rings. I*, J. Math. Soc. Japan**30**(1978), no. 2, 179–213. MR**494707**, DOI 10.2969/jmsj/03020179 - Henry B. Laufer,
*On minimally elliptic singularities*, Amer. J. Math.**99**(1977), no. 6, 1257–1295. MR**568898**, DOI 10.2307/2374025 - 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**, DOI 10.1090/S1056-3911-05-00397-8 - A. Némethi,
*Line bundles associated with normal surface singularities*, arXiv:math.AG/0310084. - András Némethi,
*“Weakly” elliptic Gorenstein singularities of surfaces*, Invent. Math.**137**(1999), no. 1, 145–167. MR**1703331**, DOI 10.1007/s002220050327 - András Némethi and Liviu I. Nicolaescu,
*Seiberg-Witten invariants and surface singularities*, Geom. Topol.**6**(2002), 269–328. MR**1914570**, DOI 10.2140/gt.2002.6.269 - A. Némethi and T. Okuma,
*On the Casson Invariant Conjecture of Neumann–Wahl*, arXiv:math.AG/0610465, to appear in J. Algebraic Geom. - Walter D. Neumann,
*Graph 3-manifolds, splice diagrams, singularities*, Singularity theory, World Sci. Publ., Hackensack, NJ, 2007, pp. 787–817. MR**2342940**, DOI 10.1142/9789812707499_{0}034 - Walter D. Neumann,
*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**, DOI 10.1090/S0002-9947-1981-0632532-8 - W. D. Neumann and J. Wahl,
*The end curve theorem for normal complex surface singularities*, arXiv:0804.4644v1. - Walter Neumann and Jonathan Wahl,
*Casson invariant of links of singularities*, Comment. Math. Helv.**65**(1990), no. 1, 58–78. MR**1036128**, DOI 10.1007/BF02566593 - Walter D. Neumann and Jonathan Wahl,
*Universal abelian covers of surface singularities*, Trends in singularities, Trends Math., Birkhäuser, Basel, 2002, pp. 181–190. MR**1900786** - Walter D. Neumann and Jonathan Wahl,
*Complete intersection singularities of splice type as universal abelian covers*, Geom. Topol.**9**(2005), 699–755. MR**2140991**, DOI 10.2140/gt.2005.9.699 - Walter D. Neumann and Jonathan Wahl,
*Complex surface singularities with integral homology sphere links*, Geom. Topol.**9**(2005), 757–811. MR**2140992**, DOI 10.2140/gt.2005.9.757 - Tomohiro Okuma,
*Universal abelian covers of rational surface singularities*, J. London Math. Soc. (2)**70**(2004), no. 2, 307–324. MR**2078895**, DOI 10.1112/S0024610704005642 - Tomohiro Okuma,
*Universal abelian covers of certain surface singularities*, Math. Ann.**334**(2006), no. 4, 753–773. MR**2209255**, DOI 10.1007/s00208-005-0693-8 - Fumio Sakai,
*Weil divisors on normal surfaces*, Duke Math. J.**51**(1984), no. 4, 877–887. MR**771385**, DOI 10.1215/S0012-7094-84-05138-X - Richard P. Stanley,
*Combinatorics and commutative algebra*, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996. MR**1453579** - Masataka Tomari and Keiichi 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. MR**1031224**, DOI 10.2977/prims/1195172704

## Additional Information

**Tomohiro Okuma**- Affiliation: Department of Education, Yamagata University, Yamagata 990-8560, Japan
- MR Author ID: 619386
- Email: okuma@e.yamagata-u.ac.jp
- 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.
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2434304