Threefolds and deformations of surface singularities. (English) Zbl 0642.14008

The authors study the deformation of the surface singularities, using the minimal model theory by Reid, Mori etc.
In § 2, a result of Laufer is generalized: Let \(f: X\to Y\) be a flat family of projective surfaces with isolated singularities only such that Y is semi-normal. Then \((i)\quad \bar K^ 2_ Y\) is lower semi- continuous; \((ii)\quad \bar K^ 2_ Y\) is locally constant if and only if f admits a simultaneous Du Val resolution; \((iii)\quad Ifp: Y\to X\) is a section and Y is connected, then f admits a weak simultaneous resolution along p if and only if the germs of \(X_ y\) along p are pairwise homeomorphic.
In § 3, a deformation of the quotient singularity is considered. The normal surface singularity is called of class T if it is a quotient singularity and it admits a \({\mathbb{Q}}\)-Gorenstein one-parameter smoothing. A P-resolution of \(X_ 0\) is a partial resolution \(g: Z_ 0\to X_ 0\) such that \(Z_ 0\) has only singularities of class T. The authors show a one-one correspondence between the components of \(Def(X_ 0)\) (the deformation space) and the P-resolutions of \(X_ 0\). In § 4, they classify the semi-canonical (or terminal) singularities without the normality assumption. This is a generalization of a result of Kawamata. - In § 5, the moduli space of surface of general type is considered. The authors show that appropriate singularities to permit on the surfaces at the boundaries of the moduli space are semi-log-canonical. They also make precise the moduli problem. - In § 7, they show that any small deformation \((X_ t,x_ t)\) of a cyclic quotient singularity is also a cyclic quotient singularity which was conjectured by Riemenschneider.
Reviewer: M.Oka


14J17 Singularities of surfaces or higher-dimensional varieties
14B07 Deformations of singularities
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
Full Text: DOI EuDML


[1] [A1] Artin, M.: On isolated rational singularities of surfaces. Am. J. Math.88, 129-136 (1966) · Zbl 0142.18602
[2] [A2] Artin, M.: Algebraic construction of Brieskorn’s resolutions. J. Algebra29, 330-348 (1974) · Zbl 0292.14013
[3] [Ben] Benveniste, X.: Sur l’anneau canonique de certaines variétés de dimension trois. Invent. Math.73, 157-164 (1983) · Zbl 0539.14025
[4] [Ber] Bertini, E.: Introduzione alla geometria proiettiva degli iperspazi. Messina 1923 · JFM 49.0484.08
[5] [Br1] Brieskorn, E.: Rationale Singularitäten komplexer Flächen, Invent. Math.4, 336-358 (1968) · Zbl 0219.14003
[6] [Br2] Brieskorn, E.: Singular elements in semi-simple algebraic groups. Proc. Int. Con. Math., Nice,2, 279-284 (1971)
[7] [D] Danilov, V.I.: Birational geometry of toric 3-folds. Math. USSR Izv.21, 269-279 (1983) · Zbl 0536.14008
[8] [E1] Elkik, R.: Singularités rationelles et déformations, Invent. Math.47, 139-147 (1978) · Zbl 0383.14005
[9] [EV] Esnault, H., Viehweg, E.: Two-dimensional quotient singularities deform to quotient singularities. Math. Ann.271, 439-449 (1985) · Zbl 0566.14002
[10] [GR] Grauert, H., Riemenschneider, O.: Verschwindungssätze für analytische Kohomolgiegrouppen auf komplexen Räumen. Invent. Math.11, 263-292 (1970) · Zbl 0202.07602
[11] [Harr] Harris, J.: A bound on the geometric genus of projective varieties. Ann. Scu. Norm. Pisa8, 35-68 (1981) · Zbl 0467.14005
[12] [Hart] Hartshorne, R.: Complete intersections and connectedness. Am. J. Math.84, 497-508 (1962) · Zbl 0108.16602
[13] [Hin] Hinic, V.A.: On the Gorenstein property of a ring of invariants. Izv. Akad. Nauk SSSR40, 50-56 (1976) (=Math USSR. Izv.10, 47-53 (1976))
[14] [I] Iitaka, S.: Birational geometry for open varieties. Sém. Math. Sup., Montreal,76 (1981) · Zbl 0491.14005
[15] [Kar1] Karras, U.: Deformations of cusp singularities. Proc. Symp. Pure Math.30, 37-44 (1970)
[16] [Kar2] Karras, U.: Weak simultaneous resolution (to appear)
[17] [Kaw1] Kawamata, Y.: On singularities in the classification theory of algebraic varieties. Math. Ann.251, 51-55 (1980) · Zbl 0437.14019
[18] [Kaw2] Kawamata, Y.: Crepant blowings-up of three-dimensional canonical singularities and its application to degenerations of surfaces. Ann. Math. (to appear)
[19] [Kaw3] Kawamata, Y.: On the finiteness of generators of a pluricanonical ring for a 3-fold of general type. Am. J. Math.106, 1503-1512 (1984) · Zbl 0587.14027
[20] [KMM] Kawamata, Y., Matsuda, K., Matsuki, K.: Introduction to the minimal model problem. In: T. Oda (ed.) Alg. Geom. Sendai Adv. Stud. Pure Math. 10 (1987) Kinokuniya-North-Holland, pp. 283-360 · Zbl 0672.14006
[21] [KKMS] Kempf, G., Knudsen, F., Mumford, D., Saint Donat, B.: Toroidal Embeddings I (Lect. Notes Math., vol. 339), Berlin Heidelberg New York: Springer 1973 · Zbl 0271.14017
[22] [Ko1] Kollár, J.: Toward moduli of singular varieties. Comp. Math.56, 369-398 (1985) · Zbl 0666.14003
[23] [Ko2] Kollár, J.: Deformations of related singularities (Preprint)
[24] [Ko3] Kollár, J.: The structure of algebraic threefolds, Bull. AMS17, 211-273 (1987) · Zbl 0649.14022
[25] [La] Laufer, H.A.: Weak simultaneous resolution of surface singularities. Proc. Symp. Pure. Math.40, part 2, 1-29 (1983)
[26] [LW] Looijenga, E., Wahl, J.: Quadratic functions and smoothing surface singularities. Topology25, 261-297 (1986) · Zbl 0615.32014
[27] [MM] Matsusaka, T., Mumford, D.: Two fundamental theorems on deformations of polarized varieties. Am. J. Math.86, 668-684 (1964) · Zbl 0128.15505
[28] [MT] Merle, M., Teissier, B.: Conditions d’adjonction, d’après DuVal (Lect. Notes Math., vol 777) Berlin Heidelberg New York: Springer 1980
[29] [Mo1] Mori, S.: On three dimensional terminal singularities. Nagoya Math. J.98, 43-66 (1985) · Zbl 0589.14005
[30] [Mo2] Mori, S.: Minimal models for semi-stable degenerations of surfaces (unpublished)
[31] [Mo3] Mori, S.: Flip conjecture and the existence of minimal models for 3-folds. J. Am. Math. Soc. (to appear)
[32] [MS] Morrison, D., Stevens, G.: Terminal quotient singularities in dimensions three and four. Proc. Am. Math. Soc.90, 15-20 (1984) · Zbl 0536.14003
[33] [M1] Mumford, D.: The topology of normal singularities of an algebraic surface and a criterion or simplicity. Pub. Math. IHES9, 5-22 (1961) · Zbl 0108.16801
[34] [M2] Mumford, D.: The stability of projective varieties, L’Ens. Math.23, 39-110 (1977)
[35] [Na] Nakayama, N.: Invariance of plurigenera under deformation. Topology25, 237-251 (1986) · Zbl 0596.14026
[36] [Ne] Neumann, W.: A calculus for plumbing and the topology of links. Trans. Am. Math. Soc.268, 299-344 (1981)
[37] [Pi1] Pinkham, H.: Deformations of algebraic varieties withG m-action. Astérisque20, (1974)
[38] [Pi2] Pinkham, H.: Simple elliptic singularities. Proc. Symp. Pure. Math.30, 69-71 (1977)
[39] [Po] Popp, H.: Moduli theory and classification theory for algebraic varieties (Lect. Notes Math., vol. 620) Berlin Heidelberg New York: Springer 1977 · Zbl 0359.14005
[40] [Re1] Reid, M.: Elliptic Gorenstein singularities of surfaces. (preprint, 1976)
[41] [Re2] Reid, M.: Canonical threefolds, Journées de Géométrie algébrique d’Angers. Sijthoff and Nordhoff (1980), pp. 273-310
[42] [Re3] Reid, M.: Minimal models of canonical threefolds, Algebraic and Analytic Varieties. Adv. Stud. Pure Math.1, 131-180 (1983)
[43] [Ri] Riemenschneider, O.: Deformation von Quotientsingularitäten (nach zyklischen Gruppen). Math. Ann.209, 211-248 (1974) · Zbl 0275.32010
[44] [Sai] Saito, K.: Einfach-elliptische Singularitäten. Invent. Math.23, 284-375 (1974) · Zbl 0296.14019
[45] [Sak] Sakai, F.: Weil divisors on normal surfaces. Duke Math. J.512, 877-888 (1984) · Zbl 0602.14006
[46] [Sal] Sally, J.: On the associated graded ring of a Cohen-Macaulay ring. J. Math. Kyoto Univ.17, 19-21 (1977) · Zbl 0353.13017
[47] [S-B1] Shepherd-Barron, N.: Some questions on singularities in two and three dimensins, Thesis, Warwick Univ., 1981 (unpublished)
[48] [S-B2] Shepherd-Barron, N.: Degenerations with numerically effective canonical divisor, The Birational Geometry of Degenerations, Progr. Math.29, 33-84 (1983)
[49] [Sh] Shokurov, V.V.: Letter to M. Reid
[50] [vS] Straten, D. van: Weakly normal surface singularities and their improvements, Thesis, Leiden, 1987
[51] [Te] Teissier, B.: Résolution simultanée I, II (Lect. Notes Math., vol. 777, pp. 71-146) Berlin Heidelberg New York: Springer 1980
[52] [Tr] Traverso, C.: Seminormality and the Picard group. Ann. Scu. Norm. Pisa75, 585-595 (1970) · Zbl 0205.50501
[53] [Ts] Tsunoda, S.: Minimal models for semi-stable degenerations of surfaces (Preprint)
[54] [TsM] Tsunoda, S., Miyanishi, M.: The structure of open algebraic surfaces II, Classification of algebraic and analytic manifolds. Progr. Math.39 (1983) · Zbl 0605.14035
[55] [V] Vaquié M.: Résolution simultanée de surfaces normales. Ann. Inst. Fourier35, 1-38 (1985)
[56] [Wa1] Wahl, J.: Equisingular deformations of normal surfaces singularities. Ann. Math.104, 325-356 · Zbl 0358.14007
[57] [Wa2] Wahl, J.: Elliptic deformations of minimally elliptic singularities. Math. Ann.253, 241-262 (1980) · Zbl 0431.14012
[58] [Wa3] Wahl, J.: Smoothing of normal surface singularities. Topology20, 219-246 (1981) · Zbl 0484.14012
[59] [X] Xambó, S.: On projective varieties of minimal degree. Collectanea Math.32, 149-163 (1981) · Zbl 0501.14020
[60] [Z] Zariski, O.: The problem of Riemann-Roch for high multiples of an effective divisor. Ann. Math.76, 560-615 (1962) · Zbl 0124.37001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.