Moduli of isolated hypersurface singularities. (English) Zbl 1084.32019

Eastwood, Michael G.

Asian J. Math. 8, No. 2, 305-314 (2004).

The author considers the problem of distinguishing between biholomorphically inequivalent hypersurface singularities on the basis of their corresponding moduli algebras. The moduli algebra of a complex hypersurface \(V\) in \(\mathbb C^n\) with an isolated singularity at the origin is defined as the quotient \(A=\mathbb C\{z_1,\dots, z_n\}/\langle f,\partial f/\partial z_1,\dots,\partial f/\partial z_n \rangle\). The author employs classical invariant theory to compute from \(A\) the \(j\)-invariant defined by K. Saito [Invent. Math. 23, 289–325 (1974; Zbl 0296.14019)]. As an application, the simple elliptic singularities \(\widetilde E_6\), \(\widetilde E_7\) and \(\widetilde E_8\) are analyzed.

Reviewer: Carles Biviá-Ausina (València)

MSC:

32S25	Complex surface and hypersurface singularities
14J17	Singularities of surfaces or higher-dimensional varieties

Keywords:

moduli algebra; elliptic singularities