Moduli of isolated hypersurface singularities (Q1769520)

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 \textit{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.