Canonical quotient singularities in dimension three
HTML articles powered by AMS MathViewer
- by David R. Morrison PDF
- Proc. Amer. Math. Soc. 93 (1985), 393-396 Request permission
Abstract:
We classify isolated canonical cyclic quotient singularities in dimension three, showing that, with two exceptions, they are all either Gorenstein or terminal. The proof uses the solution of a combinatorial problem which arose in the study of algebraic cycles on Fermat varieties.References
- Noboru Aoki, On some arithmetic problems related to the Hodge cycles on the Fermat varieties, Math. Ann. 266 (1983), no. 1, 23–54. MR 722926, DOI 10.1007/BF01458703
- Noboru Aoki and Tetsuji Shioda, Generators of the Néron-Severi group of a Fermat surface, Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser Boston, Boston, MA, 1983, pp. 1–12. MR 717587, DOI 10.1007/BF01589436
- V. I. Danilov, Birational geometry of three-dimensional toric varieties, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 971–982, 1135 (Russian). MR 675526 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 (abstract of conference reports), Moscow-Alma-Ata, 1981. (Russian)
- V. A. Hinič, When is a ring of invariants of a Gorenstein ring also a Gorenstein ring?, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 1, 50–56, 221 (Russian). MR 0424839
- Werner Meyer and Wolfram Neutsch, Fermatquadrupel, Math. Ann. 256 (1981), no. 1, 51–62 (German). MR 620122, DOI 10.1007/BF01450943
- David R. Morrison and Glenn Stevens, Terminal quotient singularities in dimensions three and four, Proc. Amer. Math. Soc. 90 (1984), no. 1, 15–20. MR 722406, DOI 10.1090/S0002-9939-1984-0722406-4
- Ziv Ran, Cycles on Fermat hypersurfaces, Compositio Math. 42 (1980/81), no. 1, 121–142. MR 594486
- Miles Reid, Canonical $3$-folds, Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, pp. 273–310. MR 605348
- Tetsuji Shioda, On the Picard number of a Fermat surface, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 725–734 (1982). MR 656049
- Yung-Sheng Tai, On the Kodaira dimension of the moduli space of abelian varieties, Invent. Math. 68 (1982), no. 3, 425–439. MR 669424, DOI 10.1007/BF01389411
- Keiichi Watanabe, Certain invariant subrings are Gorenstein. I, II, Osaka Math. J. 11 (1974), 1–8; ibid. 11 (1974), 379–388. MR 354646
- G. K. White, Lattice tetrahedra, Canadian J. Math. 16 (1964), 389–396. MR 161837, DOI 10.4153/CJM-1964-040-2
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 93 (1985), 393-396
- MSC: Primary 14B05
- DOI: https://doi.org/10.1090/S0002-9939-1985-0773987-7
- MathSciNet review: 773987