Kodaira singularities and an extension of Arnold's strange duality
Ebeling, W. ; Wall, C. T. C.
Compositio Mathematica, Tome 56 (1985), p. 3-77 / Harvested from Numdam
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@article{CM_1985__56_1_3_0,
     author = {Ebeling, Wolfgang and Wall, C. T. C.},
     title = {Kodaira singularities and an extension of Arnold's strange duality},
     journal = {Compositio Mathematica},
     volume = {56},
     year = {1985},
     pages = {3-77},
     mrnumber = {806842},
     zbl = {0586.14033},
     language = {en},
     url = {http://dml.mathdoc.fr/item/CM_1985__56_1_3_0}
}

Ebeling, W.; Wall, C. T. C. Kodaira singularities and an extension of Arnold's strange duality. Compositio Mathematica, Tome 56 (1985) pp. 3-77. http://gdmltest.u-ga.fr/item/CM_1985__56_1_3_0/

[1] A. Andreotti and T. Frankel: The Lefschetz theorem on hyperplane sections. Annals of Math. 69 (1959) 713-717. | MR 177422 | Zbl 0115.38405

[2] V.I. Arnold: Remarks on the stationary phase method and Coxeter numbers. Usp. Math. Nauk. 28 (1973) (5), 17-44(Engl. translation in Russ. Math. Surv. 28 (1973) 5, 19-48. | MR 397777 | Zbl 0291.40005

[3] V.I. Arnold: Normal forms of functions in a neighbourhood of a degenerate critical point. Usp. Math. Nauk. 29 (1974) 2, 11-49(Engl. translation in Russ. Math. Surv. 29 (1974) 2, 10-50). | MR 516034 | Zbl 0304.57018

[4] V.I. Arnold: Critical points of smooth functions and their normal forms. Usp. Math. Nauk. 30 (1975) 5, 3-65(Engl. translation in Russ. Math. Surv. 30 (1975) 5, 1-75). | MR 420689 | Zbl 0343.58001

[5] M. Artin: On isolated rational singularities of surfaces. Amer. J. Math. 88 (1966) 129-136. | MR 199191 | Zbl 0142.18602

[6] E. Brieskorn: Die Monodromie der isolierten Singularitäten von Hyperflächen. Manuscripta math. 2 (1970) 103-160. | MR 267607 | Zbl 0186.26101

[7] E. Brieskorn: Die Milnorgitter der exzeptionellen unimodularen Singularitäten. Bonner Mathematische Schriften Nr. 150, Bonn (1983). | MR 733785 | Zbl 0525.14002

[8] N. Bourbaki: Groupes et Algèbres de Lie, Chaps. 4, 5 et 6. Paris, Hermann (1968). | MR 240238

[9] D. Buchsbaum and D. Eisenbud: Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3. Amer. J. Math. 99 (1977) 447-485. | MR 453723 | Zbl 0373.13006

[10] I.V. Dolgachev: Quotient-conical singularities on complex surfaces. Funkt. Anal. Jego Prilozh. 8 (1974) 2, 75-76(Engl. translation in Funct. Anal. Appl. 8 (1974) 160-161). | Zbl 0295.14017

[11] I.V. Dolgachev: Automorphic forms and quasihomogeneous singularities. Funkt. Anal. Jego Prilozh. 9 (1975) 2, 67-68(Engl. translation in Funct. Anal. Appl. 9 (1975) 149-151). | Zbl 0321.14003

[12] I.V. Dolgachev: On the link space of a Gorenstein quasihomogeneous surface singularity. Math. Ann. 265 (1983) 529-540. | MR 721886 | Zbl 0506.14027

[13] A. Durfee: The signature of smoothings of complex surface singularities. Math. Ann. 232 (1978) 85-98. | MR 466620 | Zbl 0346.32016

[14] A. Durfee: Fifteen characterizations of rational double points and simple critical points. l'Enseignement Math. 25 (1979) 131-163. | MR 543555 | Zbl 0418.14020

[15] P. D: On the singularities which do not affect the condition of adjunction (Part I). Proc. Camb. Phil. Soc. 30 (1934) 483-491. | JFM 60.0600.01 | Zbl 0010.17701

[16] W. Ebeling: Quadratische Formen und Monodromiegruppen von Singularitäten. Math. Ann. 255 (1981) 463-498. | MR 618180 | Zbl 0438.32004

[17] W. Ebeling: Arithmetic monodromy groups. Math. Ann. 264 (1983) 241-255. | MR 711881 | Zbl 0501.14018

[18] W. Ebeling: Milnor lattices and geometric bases of some special singularities. L'Enseignement Math. 29 (1983) 263-280. | MR 719312 | Zbl 0536.14002

[19] W. Ebeling: An arithmetic characterisation of the symmetric monodromy groups of singularities. Invent. Math. 77 (1984) 85-99. | MR 751132 | Zbl 0527.14031

[20a]A.M. Gabrielov: Dynkin diagrams of unimodal singularites. Funkt. Anal. Jego Prilozh. 8 (1974) 3, 1-6(Engl. translation in Funct. Anal. Appl. 8 (1974) 192-196). | MR 367274 | Zbl 0304.14010

[20b]V.V. Goryunov: Adjoining spectra of certain singularities. Vestnik Mos. Univ. Ser. 1, Math. Mech. 4 (1981) 19-22 (in Russian). | MR 631501 | Zbl 0514.32004

[21] G.M. Greuel and J.H.M. Steenbrink: On the topology of smoothable singularities. Proc. of Symp. in Pure Math. Vol. 40, Part 1 (1983) 535-545. | MR 713090 | Zbl 0535.32004

[22] H. Hamm: Lokale topologische Eigenschaften komplexer Räume. Math. Ann. 191 (1971) 235-252. | MR 286143 | Zbl 0214.22801

[23] F. Hirzebruch: Hilbert modular surfaces. L'Enseignement Math. 19 (1973) 183-282. | MR 393045 | Zbl 0285.14007

[24] F. Hirzebruch and D. Zagier: Classification of Hilbert modular surfaces. In: Complex analysis and algebraic geometry (W.L. Baily and T. Shioda, eds.), Cambridge University Press (1977) pp. 43-77. | MR 480356 | Zbl 0354.14011

[25a]M. Inoue: New surfaces with no meromorphic functions. Proc. Internat. Cong. Math. Vancouver. Vol. 1 (1974) 423-426. | MR 442296 | Zbl 0365.14010

[25b] M. Inoue: New surfaces with no meromorphic functions II. In: Complex analysis and algebraic geometry (W.L. Baily and T. Shioda, eds.), Cambridge University Press (1977) pp. 91-106. | MR 442297 | Zbl 0365.14011

[26] V.G. Kac: Infinite root systems, representations of graphs and invariant theory. Invent. Math. 56 (1980) 57-92. | MR 557581 | Zbl 0427.17001

[27] U. Karras: Klassifikation 2-dimensionaler Singularitäten mit auflösbaren lokalen Fundamentalgruppen. Math. Ann. 213 (1975) 231-255. | MR 379894 | Zbl 0285.57019

[28] U. Karras: Eigenschaften der lokalen Ringe in zweidimensionalen Spitzen. Math. Ann. 215 (1975) 117-129. | MR 374479 | Zbl 0286.14003

[29] U. Karras: Deformations of cusp singularities. Proc. of Symp. in Pure Math. Vol. 30 (1977) 37-44. | MR 472811 | Zbl 0352.14007

[30] U. Karras: On pencils of curves and deformations of minimally elliptic singularities. Math. Ann. 247 (1980) 43-65. | MR 565138 | Zbl 0407.14014

[31] K. Kodaira: On compact analytic surfaces II. Ann. of Math. 77 (1963) 563-626. | Zbl 0118.15802

[32] V.S. Kulikov: Degenerate elliptic curves and resolution of uni- and bimodal singularities. Funkt. Anal. Jego Prilozh. 9 (1975) 1, 72-73; (Engl. translation in Funct. Anal. Appl. 9 (1975) 69-70. | MR 463490 | Zbl 0317.14013

[33] H. Laufer: Taut 2-dimensional singularities. Math. Ann. 205 (1973) 131-164. | MR 333238 | Zbl 0281.32010

[34] H. Laufer: On μ for surface singularities. Proc. of Symp. in Pure Math. Vol. 30 (1977) 45-49. | Zbl 0363.32009

[35] H. Laufer: On minimally elliptic singularities. Amer. J. Math. 99 (1977) 1257-1295. | MR 568898 | Zbl 0384.32003

[36] H. Laufer: Ambient deformations for exceptional sets in two-manifolds. Invent. Math. 55 (1979) 1-36. | MR 553993 | Zbl 0509.32010

[37] E. Looijenga: Rational surfaces with an anti-canonical cycle. Annals of Math. 114 (1981) 267-322. | MR 632841 | Zbl 0509.14035

[38] E. Looijenga: The smoothing components of a triangle singularity I. Proc. of Symp. in Pure Math. Vol. 40, Part 2 (1983) 173-183. | MR 713246 | Zbl 0568.14002

[39] E. Looijenga and J. Wahl: Quadratic functions and smoothing surface singularities. Topology (to appear). | MR 842425 | Zbl 0615.32014

[40] J.N. Mather: Stability of C∞-mappings III: Finitely determined map germs. Publ. Math. I.H.E.S. 35 (1968) 127-156. | Numdam | Zbl 0159.25001

[41] J. Milnor: Singular points of complex hypersurfaces. Ann. of Math. Studies, Princeton (1968). | MR 239612 | Zbl 0184.48405

[42] J. Milnor: On the 3-dimensional Brieskorn manifolds M( p, q, r). In: Knots, groups and 3-manifolds. Papers dedicated to the memory of R.H. Fox (L.P. Neuwirth ed.), Ann. of Math. Studies, Princeton University Press (1975) pp. 175-225. | MR 418127 | Zbl 0305.57003

[43] J.W. Morgan and D.P. Sullivan: The transversality characteristic class and linking cycles in surgery theory. Annals of Math. 99 (1974) 463-544. | MR 350748 | Zbl 0295.57008

[44] I. Nakamura: Inoue-Hirzebruch surfaces and a duality of hyperbolic unimodular singularities I. Math. Ann. 252 (1980) 221-235. | MR 593635 | Zbl 0425.14010

[45] I. Nakamura: Duality of cusp singularities. In: Complex analysis of singularities (R.I.M.S. Symposium, K. Aomoto, organizer), Kyoto (1981) pp. 1-18.

[46] W. Neumann: Geometry of quasihomogeneous surface singularities. Proc. of Symp. in Pure Math. Vol. 40, Part 2 (1983) 245-258. | MR 713253 | Zbl 0519.32011

[47] V.V. Nikulin: Integral symmetric bilinear forms and some of their applications. Izv. Akad. Nauk. SSSR Ser. Mat. 43 (1979) 111-177(Engl. translation in Math. USSR Izv. 14, No. 1 (1980) 103-167). | MR 525944 | Zbl 0427.10014

[48] H. Pinkham: Deformations of algebraic varieties with G m action. Astérisque 20, Soc. Math. de France (1974). | MR 376672 | Zbl 0304.14006

[49] H. Pinkham: Singularités exceptionelles, la dualité étrange d'Arnold et les surfaces K-3. C.R. Acad. Sc. Paris, Série A, 284 (1977) 615-618. | MR 429876 | Zbl 0375.14004

[50] H. Pinkham: Groupe de monodromie des singularités unimodulaires exceptionelles. C.R. Acad. Sc. Paris, Série A, 284 (1977) 1515-1518. | MR 439840 | Zbl 0391.14005

[51] H. Pinkham: Smoothings of the Dpqr singularities, p + q + r = 22. Appendix to Looijenga's paper: The smoothing components of a triangle singularity. Proc. of Symp. in Pure Math. Vol. 40, Part 2 (1983) 373-377. | MR 713261 | Zbl 0568.14004

[52] H. Pinkham: Automorphisms of cusps and Inoue-Hirzebruch surfaces. Compositio Math. 52 (1984) 299-313. | Numdam | MR 756724 | Zbl 0573.14015

[53] M. Reid: Elliptic Gorenstein singularities. Unpublished manuscript (1973).

[54] K. Saito: Einfach-elliptische Singularitäten. Invent. Math. 23 (1974) 289-325. | MR 354669 | Zbl 0296.14019

[55] P. Scott: The geometries of 3-manifolds. Bull. London Math. Soc. 15 (1983) 401-487. | MR 705527 | Zbl 0561.57001

[56] J.A. Seade: A cobordism invariant for surface singularities. Proc. of Symp. in Pure Math. Vol. 40, Part 2 (1983) 479-484. | MR 713273 | Zbl 0524.57016

[57] J.H.M. Steenbrink: Intersection form for quasi-homogeneous singularities. Compositio Math. 34 (1977) 211-223. | Numdam | MR 453735 | Zbl 0347.14001

[58] J.H.M. Steenbrink: Mixed Hodge structure on the vanishing cohomology. In: Real and complex singularities (P. Holm ed.), Sijthoff & Noordhoff (1977) pp. 525-563. | MR 485870 | Zbl 0373.14007

[59] J.H.M. Steenbrink: Mixed Hodge structures associated with isolated singularities. Proc. of Symp. in Pure Math. Vol. 40, Part 2 (1983) 513-536. | MR 713277 | Zbl 0515.14003

[60] W. Thurston: The geometry and topology of 3-manifolds. Notes. Princeton (1978-79).

[61] W. Thurston: Three dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. AMS 6 (1982) 357-382. | MR 648524 | Zbl 0496.57005

[62] F. Van Der Blij: An invariant of quadratic forms mod. 8. Kon. Ned. Akad. van Wet. (A) 62 (1959) 291-293. | MR 108467 | Zbl 0087.04203

[63] J. Wahl: Smoothings of normal surface singularities. Topology 20 (1981) 219-246. | MR 608599 | Zbl 0484.14012

[64] R. Waldi: Deformation von Gorenstein-Singularitäten der Kodimension 3. Math. Ann. 242 (1979) 201-208. | MR 545214 | Zbl 0426.14004

[65] C.T.C. Wall: Classification problems in differential topology-VI: Classification of (s-1)-connected (2s + 1)-manifolds. Topology 6 (1967) 273-296. | MR 216510 | Zbl 0173.26102

[66] C.T.C. Wall: Classification of unimodal isolated singularities of complete intersections. Proc. of Symp. in Pure Math. Vol. 40, Part 2 (1983) 625-640. | MR 713286 | Zbl 0519.58013

[67] C.T.C. Wall: Notes on the classification of singularities. Proc. Lond. Math. Soc. 48 (1984) 461-513. | MR 735225 | Zbl 0541.14003

[68] K. Watanabe: Certain invariant subrings are Gorenstein, I, II. Osaka J. Math. 11 (1974) 1-8, 379-388. | MR 354646 | Zbl 0292.13008