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Algebraic determination of isomorphism classes of the moduli algebras of \(\widehat E_6\) singularities. (English) Zbl 0976.32015
Let \(A = {\mathbf C}[[x,y,z]]/(f, \partial f/\partial x, \partial f/\partial y, \partial f/\partial z)\) be the moduli local algebra of an isolated hypersurface singularity \((V,0) \subset ({\mathbf C}^3, 0)\) defined by the germ of an analytic function \(f=f(x,y,z).\) Due to J. Mather and S.-T. Yau [Invent. Math. 69, 243-251 (1982; Zbl 0499.32008)] it is a finite dimensional \({\mathbf C}\)-algebra which determines the analytic type of the singularity.
The authors study the case of \(\widetilde E_6\) singularities which are defined by the one parameter family of the germs of analytic functions \(f_t(x,y,z) = x^3 + y^3 + z^3 + txyz.\) The corresponding family of moduli algebras \(A_t = {\mathbf C}[[x,y,z]]/(3x^2+tyz, 3y^2+txz, 3z^2+txy)\) can be considered as a one parameter family of commutative Artinian algebras. The set of isomorphisms of such algebras \(A_t, t^3 \neq 0, 216, -27,\) is described. It consists in fact of 216 matrices in PGL\((3, \mathbf C).\) As a consequence the following result is obtained. Let \(k(t) = t^3(t^3-216)/(t^3+27)^3.\) Then two elements \(A_t\) and \(A_s\) of this one parameter family are isomorphic if and only if \(k(t) = k(s).\) Thus, \(k(t)\) is a modulus of the family.
The authors remark that they “have enough evidence to show that Saito’s computation of the \(j\)-invariant for \(\widetilde E_6\) is in error” [cf. K. Saito, Invent. Math. 23, 289-325 (1974; Zbl 0296.14019)].

MSC:
32S15 Equisingularity (topological and analytic)
14B10 Infinitesimal methods in algebraic geometry
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
58K40 Classification; finite determinacy of map germs
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