Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On four-dimensional terminal quotient singularities

Authors: Shigefumi Mori, David R. Morrison and Ian Morrison
Journal: Math. Comp. 51 (1988), 769-786
MSC: Primary 14J35; Secondary 14B05, 14J10
MathSciNet review: 958643
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We report on an investigation of four-dimensional terminal cyclic quotient singularities which are not Gorenstein. (For simplicity, we focus on quotients by cyclic groups of prime order.) An enumeration, using a computer, of all such singularities for primes $ < 1600$ led us to conjecture a structure theorem for these singularities (which is rather more complicated than the known structure theorem in dimension three). We discuss this conjecture and our evidence for it; we also discuss properties of the anticanonical and antibicanonical linear systems of these singularities.

References [Enhancements On Off] (What's this?)

  • [1] V. I. Danilov, "The birational geometry of toric threefolds," Izv. Akad. Nauk SSSR Ser. Mat., v. 46, 1982, pp. 972-981 = Math. USSR Izv., v. 21, 1983, pp. 269-280. MR 675526 (84e:14008)
  • [2] M. A. Frumkin, Description of the Elementary Three-Dimensional Polyhedra, First All-Union Conference on Statistical and Discrete Analysis of Non-Numerical Information, Expert Estimation and Discrete Optimization (abstracts of conference reports), Moscow-Alma-Ata, 1981. (Russian)
  • [3] Y. Kawamata, "Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces," Ann. of Math., v. 127, 1988, pp. 93-163. MR 924674 (89d:14023)
  • [4] J. Kollár & N. I. Shepherd-Barron, "Threefolds and deformations of surface singularities," Invent Math., v. 91, 1988, pp. 299-338. MR 922803 (88m:14022)
  • [5] S. V. Konyagin & D. G. Markushevich, "Criteria for canonicity of cyclic quotients of regular and nondegenerate singular points," Sibirsk. Mat. Zh., v. 26, no. 4, 1985, pp. 68-78, 204 = Siberian Math. J., v. 26, 1985, pp. 530-539. MR 804019 (87b:14002)
  • [6] S. Mori, "On 3-dimensional terminal singularities," Nagoya Math. J., v. 98, 1985, pp. 43-66. MR 792770 (86m:14003)
  • [7] S. Mori, "Flip theorem and the existence of minimal models for 3-folds," J. Amer. Math. Soc., v. 1, 1988, pp. 117-253. MR 924704 (89a:14048)
  • [8] D. R. Morrison & G. Stevens, "Terminal quotient singularities in dimensions three and four," Proc. Amer. Math. Soc., v. 90, 1984, pp. 15-20. MR 722406 (85a:14004)
  • [9] W. H. Press et al., Numerical Recipes. The Art of Scientific Computing, Cambridge Univ. Press, Cambridge-New York, 1986. MR 833288 (87m:65001a)
  • [10] M. Reid, "Canonical 3-folds," in Algebraic Geometry Angers 1979 (A. Beauville, ed.), Sijthoff & Noordhoff, Alphen aan den Rijn Germantown, Md., 1980, pp. 273-310. MR 605348 (82i:14025)
  • [11] M. Reid, "Minimal models of canonical 3-folds," in Algebraic Varieties and Analytic Varieties (S. Iitaka, ed.), Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 131-180. MR 715649 (86a:14010)
  • [12] M. Reid, "Young person's guide to canonical singularities," in Algebraic Geometry (Bowdoin 1985), Proc. Sympos. Pure Math., v. 46, part 1, 1987, pp. 345-474. MR 927963 (89b:14016)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 14J35, 14B05, 14J10

Retrieve articles in all journals with MSC: 14J35, 14B05, 14J10

Additional Information

Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society