Dynkin graphs and triangle singularities. (English) Zbl 0827.14002
The author considers nine of the fourteen triangle singularities in Arnold’s list [cf. V. T. Arnold, Invent. Math. 35, 87-109 (1976; Zbl 0336.57022)]. Namely the singularities \(E_{12}\), \(E_{13}\), \(E_{14}\), \(Z_{11}\), \(Z_{12}\), \(Z_{13}\), \(Q_{10}\), \(Q_{11}\), \(Q_{12}\). He defines elementary and tie transformations on Dynkin graphs with several components. The aim of this paper is to give a description of Dynkin graphs \(\Gamma\) with several components such that there exists a small deformation fibre \(Y\) of a triangle singularity and satisfying two technical conditions, in terms of elementary and tie transformations of certain Dynkin graphs. The author recalls the definition of a root system of type \(BC\), \(G\) and also explains why this method does not apply to the study of the remaining triangle singularities, namely \(W_{12}\), \(W_{13}\), \(S_{11}\), \(S_{12}\) and \(U_{12}\). He announces that some results on these singularities will appear elsewhere.
Reviewer: M.Morales (Saint-Martin-d’Heres)
MSC:
14B05 | Singularities in algebraic geometry |
14J17 | Singularities of surfaces or higher-dimensional varieties |
32S30 | Deformations of complex singularities; vanishing cycles |
14B12 | Local deformation theory, Artin approximation, etc. |
Citations:
Zbl 0336.57022References:
[1] | Arnold, V. : Local normal forms of functions. Invent. Math. 35. 87-109 (1976). · Zbl 0336.57022 · doi:10.1007/BF01390134 |
[2] | Bourbaki, N. . Groupes et algebre de Lie. Chaps. 4-6, Paris: Hermann 1968 · Zbl 0186.33001 |
[3] | Gabrielov, A. M. : Dynkin diagrams for unimodular singularities. Funkt. Anal. Jego Prilozh. 8:3 1-6 (1974). · Zbl 0304.14010 · doi:10.1007/BF01075691 |
[4] | Looijenga, E. : The smoothing components ofa triangle singularity. II. Math. Ann. 269. 357-38 (1984). · Zbl 0568.14003 · doi:10.1007/BF01450700 |
[5] | Nikulin, V. V. : Integral symmetric bilinear forms and some of their applications. Mat. USS Izv. 43 No. 1, (1979) (English translation: Math. USSR Izv. 14 No. 1, 103-167 (1980)). · Zbl 0408.10011 |
[6] | Urabe, T. : Dynkin graphs and quadrilateral singularities. Lecture Notes in Mathematics 1548 Springer 1993. · Zbl 0798.14020 |
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