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Simple elliptic hypersurface singularities: a new look at the equivalence problem. (English) Zbl 1228.32029
Fukui, Toshizumi (ed.) et al., The Japanese-Australian workshop on real and complex singularities, JARCS III, The University of Sydney, Sydney, Australia, September 15–18, 2009. Canberra: Australian National University, Centre for Mathematics and its Applications (ISBN 0-7315-5207-5). Proceedings of the Centre for Mathematics and its Applications, Australian National University 43, 9-17 (2010).
Summary: Let \(V_1\), \(V_2\) be hypersurface germs in \(\mathbb C^m\), with \(m \geq 2\), each having a quasi-homogeneous isolated singularity at the origin. In our recent joint article with G. Fels, W. Kaup and N. Kruzhilin [J. Geom. Anal. 21, No. 3, 767–782 (2011; Zbl 1274.32018)], we reduced the biholomorphic equivalence problem for \(V_1\), \(V_2\) to verifying whether certain polynomials arising from the moduli algebras of \(V_1\), \(V_2\) are equivalent up to scale by means of a linear transformation. In the present note we illustrate this result by the examples of simple elliptic singularities of types \(\widetilde E_6\), \(\widetilde E_7\), \(\widetilde E_8\) and compare our method with that due to M. G. Eastwood who has also introduced certain polynomials that distinguish non-equivalent singularities within each of these three types.
For the entire collection see [Zbl 1218.14001].
32S25 Complex surface and hypersurface singularities
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)