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Threefolds and deformations of surface singularities

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References

  • [A1] Artin, M.: On isolated rational singularities of surfaces. Am. J. Math.88, 129–136 (1966)

    Google Scholar 

  • [A2] Artin, M.: Algebraic construction of Brieskorn's resolutions. J. Algebra29, 330–348 (1974)

    Google Scholar 

  • [Ben] Benveniste, X.: Sur l'anneau canonique de certaines variétés de dimension trois. Invent. Math.73, 157–164 (1983)

    Google Scholar 

  • [Ber] Bertini, E.: Introduzione alla geometria proiettiva degli iperspazi. Messina 1923

  • [Br1] Brieskorn, E.: Rationale Singularitäten komplexer Flächen, Invent. Math.4, 336–358 (1968)

    Google Scholar 

  • [Br2] Brieskorn, E.: Singular elements in semi-simple algebraic groups. Proc. Int. Con. Math., Nice,2, 279–284 (1971)

    Google Scholar 

  • [D] Danilov, V.I.: Birational geometry of toric 3-folds. Math. USSR Izv.21, 269–279 (1983)

    Google Scholar 

  • [E1] Elkik, R.: Singularités rationelles et déformations, Invent. Math.47, 139–147 (1978)

    Google Scholar 

  • [EV] Esnault, H., Viehweg, E.: Two-dimensional quotient singularities deform to quotient singularities. Math. Ann.271, 439–449 (1985)

    Google Scholar 

  • [GR] Grauert, H., Riemenschneider, O.: Verschwindungssätze für analytische Kohomolgiegrouppen auf komplexen Räumen. Invent. Math.11, 263–292 (1970)

    Google Scholar 

  • [Harr] Harris, J.: A bound on the geometric genus of projective varieties. Ann. Scu. Norm. Pisa8, 35–68 (1981)

    Google Scholar 

  • [Hart] Hartshorne, R.: Complete intersections and connectedness. Am. J. Math.84, 497–508 (1962)

    Google Scholar 

  • [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))

    Google Scholar 

  • [I] Iitaka, S.: Birational geometry for open varieties. Sém. Math. Sup., Montreal,76 (1981)

  • [Kar1] Karras, U.: Deformations of cusp singularities. Proc. Symp. Pure Math.30, 37–44 (1970)

    Google Scholar 

  • [Kar2] Karras, U.: Weak simultaneous resolution (to appear)

  • [Kaw1] Kawamata, Y.: On singularities in the classification theory of algebraic varieties. Math. Ann.251, 51–55 (1980)

    Google Scholar 

  • [Kaw2] Kawamata, Y.: Crepant blowings-up of three-dimensional canonical singularities and its application to degenerations of surfaces. Ann. Math. (to appear)

  • [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)

    Google Scholar 

  • [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

  • [KKMS] Kempf, G., Knudsen, F., Mumford, D., Saint Donat, B.: Toroidal Embeddings I (Lect. Notes Math., vol. 339), Berlin Heidelberg New York: Springer 1973

    Google Scholar 

  • [Ko1] Kollár, J.: Toward moduli of singular varieties. Comp. Math.56, 369–398 (1985)

    Google Scholar 

  • [Ko2] Kollár, J.: Deformations of related singularities (Preprint)

  • [Ko3] Kollár, J.: The structure of algebraic threefolds, Bull. AMS17, 211–273 (1987)

    Google Scholar 

  • [La] Laufer, H.A.: Weak simultaneous resolution of surface singularities. Proc. Symp. Pure. Math.40, part 2, 1–29 (1983)

    Google Scholar 

  • [LW] Looijenga, E., Wahl, J.: Quadratic functions and smoothing surface singularities. Topology25, 261–297 (1986)

    Google Scholar 

  • [MM] Matsusaka, T., Mumford, D.: Two fundamental theorems on deformations of polarized varieties. Am. J. Math.86, 668–684 (1964)

    Google Scholar 

  • [MT] Merle, M., Teissier, B.: Conditions d'adjonction, d'après DuVal (Lect. Notes Math., vol 777) Berlin Heidelberg New York: Springer 1980

    Google Scholar 

  • [Mo1] Mori, S.: On three dimensional terminal singularities. Nagoya Math. J.98, 43–66 (1985)

    Google Scholar 

  • [Mo2] Mori, S.: Minimal models for semi-stable degenerations of surfaces (unpublished)

  • [Mo3] Mori, S.: Flip conjecture and the existence of minimal models for 3-folds. J. Am. Math. Soc. (to appear)

  • [MS] Morrison, D., Stevens, G.: Terminal quotient singularities in dimensions three and four. Proc. Am. Math. Soc.90, 15–20 (1984)

    Google Scholar 

  • [M1] Mumford, D.: The topology of normal singularities of an algebraic surface and a criterion or simplicity. Pub. Math. IHES9, 5–22 (1961)

    Google Scholar 

  • [M2] Mumford, D.: The stability of projective varieties, L'Ens. Math.23, 39–110 (1977)

    Google Scholar 

  • [Na] Nakayama, N.: Invariance of plurigenera under deformation. Topology25, 237–251 (1986)

    Google Scholar 

  • [Ne] Neumann, W.: A calculus for plumbing and the topology of links. Trans. Am. Math. Soc.268, 299–344 (1981)

    Google Scholar 

  • [Pi1] Pinkham, H.: Deformations of algebraic varieties withG m-action. Astérisque20, (1974)

  • [Pi2] Pinkham, H.: Simple elliptic singularities. Proc. Symp. Pure. Math.30, 69–71 (1977)

    Google Scholar 

  • [Po] Popp, H.: Moduli theory and classification theory for algebraic varieties (Lect. Notes Math., vol. 620) Berlin Heidelberg New York: Springer 1977

    Google Scholar 

  • [Re1] Reid, M.: Elliptic Gorenstein singularities of surfaces. (preprint, 1976)

  • [Re2] Reid, M.: Canonical threefolds, Journées de Géométrie algébrique d'Angers. Sijthoff and Nordhoff (1980), pp. 273–310

  • [Re3] Reid, M.: Minimal models of canonical threefolds, Algebraic and Analytic Varieties. Adv. Stud. Pure Math.1, 131–180 (1983)

    Google Scholar 

  • [Ri] Riemenschneider, O.: Deformation von Quotientsingularitäten (nach zyklischen Gruppen). Math. Ann.209, 211–248 (1974)

    Google Scholar 

  • [Sai] Saito, K.: Einfach-elliptische Singularitäten. Invent. Math.23, 284–375 (1974)

    Google Scholar 

  • [Sak] Sakai, F.: Weil divisors on normal surfaces. Duke Math. J.512, 877–888 (1984)

    Google Scholar 

  • [Sal] Sally, J.: On the associated graded ring of a Cohen-Macaulay ring. J. Math. Kyoto Univ.17, 19–21 (1977)

    Google Scholar 

  • [S-B1] Shepherd-Barron, N.: Some questions on singularities in two and three dimensins, Thesis, Warwick Univ., 1981 (unpublished)

  • [S-B2] Shepherd-Barron, N.: Degenerations with numerically effective canonical divisor, The Birational Geometry of Degenerations, Progr. Math.29, 33–84 (1983)

    Google Scholar 

  • [Sh] Shokurov, V.V.: Letter to M. Reid

  • [vS] Straten, D. van: Weakly normal surface singularities and their improvements, Thesis, Leiden, 1987

  • [Te] Teissier, B.: Résolution simultanée I, II (Lect. Notes Math., vol. 777, pp. 71–146) Berlin Heidelberg New York: Springer 1980

    Google Scholar 

  • [Tr] Traverso, C.: Seminormality and the Picard group. Ann. Scu. Norm. Pisa75, 585–595 (1970)

    Google Scholar 

  • [Ts] Tsunoda, S.: Minimal models for semi-stable degenerations of surfaces (Preprint)

  • [TsM] Tsunoda, S., Miyanishi, M.: The structure of open algebraic surfaces II, Classification of algebraic and analytic manifolds. Progr. Math.39 (1983)

  • [V] Vaquié M.: Résolution simultanée de surfaces normales. Ann. Inst. Fourier35, 1–38 (1985)

    Google Scholar 

  • [Wa1] Wahl, J.: Equisingular deformations of normal surfaces singularities. Ann. Math.104, 325–356

  • [Wa2] Wahl, J.: Elliptic deformations of minimally elliptic singularities. Math. Ann.253, 241–262 (1980)

    Google Scholar 

  • [Wa3] Wahl, J.: Smoothing of normal surface singularities. Topology20, 219–246 (1981)

    Google Scholar 

  • [X] Xambó, S.: On projective varieties of minimal degree. Collectanea Math.32, 149–163 (1981)

    Google Scholar 

  • [Z] Zariski, O.: The problem of Riemann-Roch for high multiples of an effective divisor. Ann. Math.76, 560–615 (1962)

    Google Scholar 

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Kollár, J., Shepherd-Barron, N.I. Threefolds and deformations of surface singularities. Invent Math 91, 299–338 (1988). https://doi.org/10.1007/BF01389370

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