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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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