Kodaira singularities and an extension of Arnold's strange duality

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1. [1] A. Andreotti and T. Frankel: The Lefschetz theorem on hyperplane sections. Annals of Math.69 (1959) 713-717. Zbl0115.38405MR177422
2. [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. Zbl0291.40005MR397777
3. [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). Zbl0304.57018MR516034
4. [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). Zbl0343.58001MR420689
5. [5] M. Artin: On isolated rational singularities of surfaces. Amer. J. Math.88 (1966) 129-136. Zbl0142.18602MR199191
6. [6] E. Brieskorn: Die Monodromie der isolierten Singularitäten von Hyperflächen. Manuscripta math.2 (1970) 103-160. Zbl0186.26101MR267607
7. [7] E. Brieskorn: Die Milnorgitter der exzeptionellen unimodularen Singularitäten. Bonner Mathematische Schriften Nr. 150, Bonn (1983). Zbl0525.14002MR733785
8. [8] N. Bourbaki: Groupes et Algèbres de Lie, Chaps. 4, 5 et 6. Paris, Hermann (1968). MR240238
9. [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. Zbl0373.13006MR453723
10. [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). Zbl0295.14017
11. [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). Zbl0321.14003
12. [12] I.V. Dolgachev: On the link space of a Gorenstein quasihomogeneous surface singularity. Math. Ann.265 (1983) 529-540. Zbl0506.14027MR721886
13. [13] A. Durfee: The signature of smoothings of complex surface singularities. Math. Ann.232 (1978) 85-98. Zbl0346.32016MR466620
14. [14] A. Durfee: Fifteen characterizations of rational double points and simple critical points. l'Enseignement Math.25 (1979) 131-163. Zbl0418.14020MR543555
15. [15] P. D: On the singularities which do not affect the condition of adjunction (Part I). Proc. Camb. Phil. Soc.30 (1934) 483-491. Zbl0010.17701JFM60.0600.01
16. [16] W. Ebeling: Quadratische Formen und Monodromiegruppen von Singularitäten. Math. Ann.255 (1981) 463-498. Zbl0438.32004MR618180
17. [17] W. Ebeling: Arithmetic monodromy groups. Math. Ann.264 (1983) 241-255. Zbl0501.14018MR711881
18. [18] W. Ebeling: Milnor lattices and geometric bases of some special singularities. L'Enseignement Math.29 (1983) 263-280. Zbl0536.14002MR719312
19. [19] W. Ebeling: An arithmetic characterisation of the symmetric monodromy groups of singularities. Invent. Math.77 (1984) 85-99. Zbl0527.14031MR751132
20. [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). Zbl0304.14010MR367274
21. [20b] V.V. Goryunov: Adjoining spectra of certain singularities. Vestnik Mos. Univ. Ser.1, Math. Mech.4 (1981) 19-22 (in Russian). Zbl0514.32004MR631501
22. [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. Zbl0535.32004MR713090
23. [22] H. Hamm: Lokale topologische Eigenschaften komplexer Räume. Math. Ann.191 (1971) 235-252. Zbl0214.22801MR286143
24. [23] F. Hirzebruch: Hilbert modular surfaces. L'Enseignement Math.19 (1973) 183-282. Zbl0285.14007MR393045
25. [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. Zbl0354.14011MR480356
26. [25a] M. Inoue: New surfaces with no meromorphic functions. Proc. Internat. Cong. Math. Vancouver. Vol.1 (1974) 423-426. Zbl0365.14010MR442296
27. [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. Zbl0365.14011MR442297
28. [26] V.G. Kac: Infinite root systems, representations of graphs and invariant theory. Invent. Math.56 (1980) 57-92. Zbl0427.17001MR557581
29. [27] U. Karras: Klassifikation 2-dimensionaler Singularitäten mit auflösbaren lokalen Fundamentalgruppen. Math. Ann.213 (1975) 231-255. Zbl0285.57019MR379894
30. [28] U. Karras: Eigenschaften der lokalen Ringe in zweidimensionalen Spitzen. Math. Ann.215 (1975) 117-129. Zbl0286.14003MR374479
31. [29] U. Karras: Deformations of cusp singularities. Proc. of Symp. in Pure Math. Vol.30 (1977) 37-44. Zbl0352.14007MR472811
32. [30] U. Karras: On pencils of curves and deformations of minimally elliptic singularities. Math. Ann.247 (1980) 43-65. Zbl0407.14014MR565138
33. [31] K. Kodaira: On compact analytic surfaces II. Ann. of Math.77 (1963) 563-626. Zbl0118.15802
34. [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. Zbl0317.14013MR463490
35. [33] H. Laufer: Taut 2-dimensional singularities. Math. Ann.205 (1973) 131-164. Zbl0281.32010MR333238
36. [34] H. Laufer: On μ for surface singularities. Proc. of Symp. in Pure Math. Vol.30 (1977) 45-49. Zbl0363.32009
37. [35] H. Laufer: On minimally elliptic singularities. Amer. J. Math.99 (1977) 1257-1295. Zbl0384.32003MR568898
38. [36] H. Laufer: Ambient deformations for exceptional sets in two-manifolds. Invent. Math.55 (1979) 1-36. Zbl0509.32010MR553993
39. [37] E. Looijenga: Rational surfaces with an anti-canonical cycle. Annals of Math.114 (1981) 267-322. Zbl0509.14035MR632841
40. [38] E. Looijenga: The smoothing components of a triangle singularity I. Proc. of Symp. in Pure Math. Vol. 40, Part 2 (1983) 173-183. Zbl0568.14002MR713246
41. [39] E. Looijenga and J. Wahl: Quadratic functions and smoothing surface singularities. Topology (to appear). Zbl0615.32014MR842425
42. [40] J.N. Mather: Stability of C∞-mappings III: Finitely determined map germs. Publ. Math. I.H.E.S.35 (1968) 127-156. Zbl0159.25001
43. [41] J. Milnor: Singular points of complex hypersurfaces. Ann. of Math. Studies, Princeton (1968). Zbl0184.48405MR239612
44. [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. Zbl0305.57003MR418127
45. [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. Zbl0295.57008MR350748
46. [44] I. Nakamura: Inoue-Hirzebruch surfaces and a duality of hyperbolic unimodular singularities I. Math. Ann.252 (1980) 221-235. Zbl0425.14010MR593635
47. [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. 
48. [46] W. Neumann: Geometry of quasihomogeneous surface singularities. Proc. of Symp. in Pure Math. Vol. 40, Part 2 (1983) 245-258. Zbl0519.32011MR713253
49. [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). Zbl0427.10014MR525944
50. [48] H. Pinkham: Deformations of algebraic varieties with G m action. Astérisque20, Soc. Math. de France (1974). Zbl0304.14006MR376672
51. [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. Zbl0375.14004MR429876
52. [50] H. Pinkham: Groupe de monodromie des singularités unimodulaires exceptionelles. C.R. Acad. Sc. Paris, Série A, 284 (1977) 1515-1518. Zbl0391.14005MR439840
53. [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. Zbl0568.14004MR713261
54. [52] H. Pinkham: Automorphisms of cusps and Inoue-Hirzebruch surfaces. Compositio Math.52 (1984) 299-313. Zbl0573.14015MR756724
55. [53] M. Reid: Elliptic Gorenstein singularities. Unpublished manuscript (1973). 
56. [54] K. Saito: Einfach-elliptische Singularitäten. Invent. Math.23 (1974) 289-325. Zbl0296.14019MR354669
57. [55] P. Scott: The geometries of 3-manifolds. Bull. London Math. Soc.15 (1983) 401-487. Zbl0561.57001MR705527
58. [56] J.A. Seade: A cobordism invariant for surface singularities. Proc. of Symp. in Pure Math. Vol. 40, Part 2 (1983) 479-484. Zbl0524.57016MR713273
59. [57] J.H.M. Steenbrink: Intersection form for quasi-homogeneous singularities. Compositio Math.34 (1977) 211-223. Zbl0347.14001MR453735
60. [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. Zbl0373.14007MR485870
61. [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. Zbl0515.14003MR713277
62. [60] W. Thurston: The geometry and topology of 3-manifolds. Notes. Princeton (1978-79). 
63. [61] W. Thurston: Three dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. AMS6 (1982) 357-382. Zbl0496.57005MR648524
64. [62] F. Van Der Blij: An invariant of quadratic forms mod. 8. Kon. Ned. Akad. van Wet. (A) 62 (1959) 291-293. Zbl0087.04203MR108467
65. [63] J. Wahl: Smoothings of normal surface singularities. Topology20 (1981) 219-246. Zbl0484.14012MR608599
66. [64] R. Waldi: Deformation von Gorenstein-Singularitäten der Kodimension 3. Math. Ann.242 (1979) 201-208. Zbl0426.14004MR545214
67. [65] C.T.C. Wall: Classification problems in differential topology-VI: Classification of (s-1)-connected (2s + 1)-manifolds. Topology6 (1967) 273-296. Zbl0173.26102MR216510
68. [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. Zbl0519.58013MR713286
69. [67] C.T.C. Wall: Notes on the classification of singularities. Proc. Lond. Math. Soc.48 (1984) 461-513. Zbl0541.14003MR735225
70. [68] K. Watanabe: Certain invariant subrings are Gorenstein, I, II. Osaka J. Math.11 (1974) 1-8, 379-388. Zbl0292.13008MR354646

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