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A note on resolution of rational and hypersurface singularities

Author: D. A. Stepanov
Journal: Proc. Amer. Math. Soc. 136 (2008), 2647-2654
MSC (2000): Primary 14B05; Secondary 32S50
Published electronically: April 11, 2008
Erratum: Proc. Amer. Math. Soc. 138 (2010), 3019-3020
MathSciNet review: 2399025
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Abstract: It is well known that the exceptional set in a resolution of a rational surface singularity is a tree of rational curves. We generalize the combinatoric part of this statement to higher dimensions and show that the highest cohomologies of the dual complex associated to a resolution of an isolated rational singularity vanish. We also prove that the dual complex associated to a resolution of an isolated hypersurface singularity is simply connected. As a consequence, we show that the dual complex associated to a resolution of a 3-dimensional Gorenstein terminal singularity has the homotopy type of a point.

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  • 1. Abramovich, D., Karu, K., Matsuki, K., Włodarczyk, J. Torification and factorization of birational maps, J. Amer. Math. Soc. 15 (2002), 531-572. MR 1896232 (2003c:14016)
  • 2. A'Campo, N. La fonction zeta d'une monodromie, Comment. Math. Helvetici 50 (1975), 233-248. MR 0371889 (51:8106)
  • 3. Artin, M. On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129-136. MR 0199191 (33:7340)
  • 4. Dold, A. Lectures on Algebraic Topology, Springer-Verlag, 1972. MR 0415602 (54:3685)
  • 5. Eisenbud, D. Commutative Algebra with a View toward Algebraic Geometry, Springer-Verlag, 1995. MR 1322960 (97a:13001)
  • 6. Elkik, R. Rationalité des singularités canoniques, Invent. Math. 64 (1981), 1-6. MR 621766 (83a:14003)
  • 7. Gordon, G.L. On a simplicial complex associated to the monodromy, Transactions of the AMS 261 (1980), 93-101. MR 576865 (81j:32017)
  • 8. Gordon, G.L. On the degeneracy of a spectral sequence associated to normal crossings, Pacific J. Math. 90(2) (1980), 389-396. MR 600638 (83f:32011)
  • 9. Grauert, H., et al. Several complex variables VII. Sheaf-theoretical methods in complex analysis, Encyclopedia of Mathematical Sciences, 74, Springer-Verlag, Berlin, 1994. MR 1326617 (96k:32001)
  • 10. Griffiths, P., Harris, J. Principles of Algebraic Geometry, v. 1, John Wiley and Sons, New York, 1978. MR 507725 (80b:14001)
  • 11. Grothendieck, A. Éléments de géométrie algébrique, Ch. III, Institut des Hautes Études Scientifiques, Publications Mathématiques, No. 11 (1961). MR 0217085 (36:177c)
  • 12. Hironaka, H. Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Annals of Math. 79 (1964), 109-326. MR 0199184 (33:7333)
  • 13. Kulikov, Vik.S., Kurchanov, P.F. Complex algebraic varieties: periods of integrals and Hodge structures, Algebraic Geometry, III, 1-217. Encyclopaedia Math. Sci., 36, Springer, Berlin, 1998. MR 1602375
  • 14. Milnor, J. Singular Points of Complex Hypersurfaces, Annals of Mathematics Studies, 61, Princeton University Press, Princeton, NJ, 1968. MR 0239612 (39:969)
  • 15. Reid, M. Minimal Models of Canonical $ 3$-folds, Algebraic Varieties and Analytic Varieties, Adv. Studies in Pure Math., Kinokuniya and North-Holland (1983), V. 1, 131-180. MR 715649 (86a:14010)
  • 16. Stepanov, D.A. A note on the dual complex associated to a resolution of singularities (Russian), Uspekhi Matem. Nauk 61(1) (2006), 185-186. MR 2239783 (2007c:14010)

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Additional Information

D. A. Stepanov
Affiliation: Department of Mathematical Modeling, Bauman Moscow State Technical University, Moscow 105005, Russia

Keywords: Rational singularity, hypersurface singularity, resolution of singularities, the dual complex associated to a resolution
Received by editor(s): March 20, 2006
Received by editor(s) in revised form: July 2, 2006, and November 16, 2006
Published electronically: April 11, 2008
Additional Notes: This research was supported by RFBR, grant no. 05-01-00353, CRDF, grant no. RUM1-2692-MO-05, and the Program for the Development of Scientific Potential of the High School, no.
Communicated by: Ted Chinburg
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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