Terminal quotient singularities in dimensions three and four. (English) Zbl 0536.14003
M. Reid gave a criterion saying which cyclic quotient singularities are terminal singularities. - The authors use a combinatorial lemma - due to G. K. White and reproved in this note - to interpret this criterion in certain cases and they obtain thereby a complete description of isolated terminal cyclic quotient singularities in dimension three and of isolated Gorenstein terminal cyclic quotient singularities in dimension four.
Reviewer: E.Viehweg
MSC:
| 14B05 | Singularities in algebraic geometry |
| 32S05 | Local complex singularities |
| 14L30 | Group actions on varieties or schemes (quotients) |
| 11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
Keywords:
canonical singularities; Bernoulli functions; Gorenstein singularities; cyclic quotient singularities; terminal singularitiesReferences:
| [1] | Claude Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778 – 782. · Zbl 0065.26103 · doi:10.2307/2372597 |
| [2] | V. I. Danilov, Birational geometry of three-dimensional toric varieties, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 971 – 982, 1135 (Russian). |
| [3] | M. A. Frumkin, Description of elementary three-dimensional polyhedra, First All-Union Conference on Statistical and Discrete Analysis of Non-Numerical Information, Experimental Bounds and Discrete Optimization, Abstract of Conference Reports, Moscow-Alma-Ata, 1981. (Russian) |
| [4] | Akira Fujiki, On resolutions of cyclic quotient singularities, Publ. Res. Inst. Math. Sci. 10 (1974/75), no. 1, 293 – 328. · Zbl 0313.32012 · doi:10.2977/prims/1195192183 |
| [5] | Kenkichi Iwasawa, Lectures on \?-adic \?-functions, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. Annals of Mathematics Studies, No. 74. · Zbl 0236.12001 |
| [6] | 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). · Zbl 0341.13010 |
| [7] | Daniel S. Kubert and Serge Lang, Modular units, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 244, Springer-Verlag, New York-Berlin, 1981. · Zbl 0492.12002 |
| [8] | Shigefumi Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), no. 1, 133 – 176. · Zbl 0557.14021 · doi:10.2307/2007050 |
| [9] | David Prill, Local classification of quotients of complex manifolds by discontinuous groups, Duke Math. J. 34 (1967), 375 – 386. · Zbl 0179.12301 |
| [10] | 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. |
| [11] | -, Minimal models of canonical \( 3\)-folds, Algebriac Varieties and Analytic Varieties , Advanced Studies in Pure Math., vol. 1, North-Holland, Amsterdam, 1983. · Zbl 0558.14028 |
| [12] | Michael Artin and John Tate , Arithmetic and geometry. Vol. I, Progress in Mathematics, vol. 35, Birkhäuser, Boston, Mass., 1983. Arithmetic; Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday. Michael Artin and John Tate , Arithmetic and geometry. Vol. II, Progress in Mathematics, vol. 36, Birkhäuser, Boston, Mass., 1983. Geometry; Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday. · Zbl 0518.00004 |
| [13] | G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canadian J. Math. 6 (1954), 274 – 304. · Zbl 0055.14305 |
| [14] | Yung-Sheng Tai, On the Kodaira dimension of the moduli space of abelian varieties, Invent. Math. 68 (1982), no. 3, 425 – 439. · Zbl 0508.14038 · doi:10.1007/BF01389411 |
| [15] | S. Tsunoda, Degeneration of minimal surfaces with non-negative Kodaira dimension, Proc. Sympos. Algebraic Geometry, Kinosaki, Japan, 1981. (Japanese) |
| [16] | Keiichi Watanabe, Certain invariant subrings are Gorenstein. I, II, Osaka J. Math. 11 (1974), 1 – 8; ibid. 11 (1974), 379 – 388. · Zbl 0292.13008 |
| [17] | G. K. White, Lattice tetrahedra, Canad. J. Math. 16 (1964), 389 – 396. · Zbl 0124.02901 · doi:10.4153/CJM-1964-040-2 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
