Canonical quotient singularities in dimension three

Morrison, David R.

doi:10.1090/S0002-9939-1985-0773987-7

Canonical quotient singularities in dimension three
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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

Description of the elementary three-dimensional polyhedra

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