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

DOI:
https://doi.org/10.1090/S0025-5718-1988-0958643-5

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 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.

**[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)**

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

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

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1988-0958643-5

Article copyright:
© Copyright 1988
American Mathematical Society