Abstract
Every normal complex surface singularity with 
-homology sphere link has a universal abelian cover. It has been conjectured by Neumann and Wahl that the universal abelian cover of a rational or minimally elliptic singularity is a complete intersection singularity defined by a system of ``splice diagram equations''. In this paper we introduce a Neumann-Wahl system, which is an analogue of the system of splice diagram equations, and prove the following.
If (X, o) is a rational or minimally elliptic singularity, then its universal abelian cover (Y, o) is an equisingular deformation of an isolated complete intersection singularity (Y 0, o) defined by a Neumann-Wahl system. Furthermore, if G denotes the Galois group of the covering Y → X, then G also acts on Y 0 and X is an equisingular deformation of the quotient Y 0/G.
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References
Brieskorn, E.: Singular elements of semi-simple algebraic groups, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, Gauthier-Villars, Paris, 1971, pp. 279–284
Ishii, S.: Simultaneous canonical models of deformations of isolated singularities, Algebraic geometry and analytic geometry (Tokyo, 1990), ICM-90 Satell. Conf. Proc., Springer, Tokyo, 1991, pp. 81–100
Laufer, H.: On rational singularities, Amer. J. Math. 94, 597–608 (1972)
Laufer, H.: On minimally elliptic singularities, Amer. J. Math. 99, 1257–1295 (1977)
Laufer, H.: Weak simultaneous resolution for deformations of Gorenstein surface singularities, Singularities, Part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, R.I., 1983, pp. 1–29
Neumann, W.D.: A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Amer. Math. Soc. 268, 299–344 (1981)
Neumann, W.D.: Abelian covers of quasihomogeneous surface singularities, Singularities, Part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., vol.40, Amer. Math. Soc., Providence, RI, 1983, pp.233–243
Neumann, W.D., Wahl, J.: Universal abelian covers of surface singularities, Trends in singularities, Trends Math., Birkhäuser, Basel, 2002, pp.181–190
Neumann, W.D., Wahl, J.:Complex surface singularities with integral homology sphere links, 2003, preprint (math.AG/0301165)
Neumann, W.D., Wahl, J.: Universal abelian covers of quotient-cusps, Math. Ann. 326, 75–93 (2003)
Neumann, W.D., Wahl, J.: Complete intersection singularities of splice type as universal abelian covers, 2004, preprint (math.AG/0407287)
Okuma, T.: Universal abelian covers of rational surface singularities, J. London Math. Soc. (2) 70, 307–324 (2004)
Reid, M.: Chapters on algebraic surfaces, Complex algebraic geometry, IAS/Park City Math. Ser., vol.3, Amer. Math. Soc., Providence, RI, 1997, pp. 3–159
Tomari, M., Watanabe, K.-i.: Filtered rings, filtered blowing-ups and normal two- dimensional singularities with ``star-shaped'' resolution, Publ. Res. Inst. Math. Sci. 25, 681–740 (1989)
Wahl, J.: Equisingular deformations of normal surface singularities, I, Ann. of Math. 104, 325–365 (1976)
Wahl, J.: Deformations of quasi-homogeneous surface singularities, Math. Ann. 280, 105–128 (1988)
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Dedicated to Professor Jonathan Wahl on his sixtieth birthday.
This research was partially supported by the Grant-in-Aid for Young Scientists (B), The Ministry of Education, Culture, Sports, Science and Technology, Japan.