The versal deformation of the \(E_ 6\)-singularity and a family of cubic surfaces
Consider two families of cubic surfaces with homogeneous coordinates \((x : y : z : w)\) in \(\mathbb{P}^3\). The first is given by the equation \[ x^3 - 2yz^2 - y^2w + x(s_1 w^2 + s_2zw + s_3 z^2) + s_4 w^3 + s_5 zw^2 + s_6 z^2w = 0 \] over the six- dimensional parameter space \(S\) with coordinates \(s_i\), \(i = 1, \ldots, 6\). It is not difficult to see that the restriction of the family to the hyperplane \(\{w = 1\}\) is nothing but the minimal versal deformation of the simple \(E_6\)-singularity whose normal form \(f = x^3 + y^4 + z^2\) can be rewritten also as \(f' = x^3 - 2yz^2 - y^2\) by means of an elementary change of variables. -- The second family is defined by the equation \[ \begin{multlined} \rho w \{\lambda x^2 + \mu y^2 + \nu z^2 + (\rho - 1)^2 (\lambda \mu \nu \rho - 1)^2 w^2 +\\ + (\mu \nu + 1) yz + (\lambda \nu + 1) xz + (\lambda \mu + 1) xy -\\ - (\rho - 1) (\lambda \mu \nu \rho - 1) w [(\lambda + 1) x + (\mu + 1) y + (\nu + 1) z]\} + xyz = 0\end{multlined} \] over the four-dimensional parameter space \(\Lambda\) with coordinates \(\lambda, \mu, \nu, \rho\). In fact, this family is a modified Cayley family [see \textit{I. Naruki} and the author, Proc. Jap. Acad., Ser. A 56, 122-125 (1980; Zbl 0472.14018)] containing all nonsingular cubics in \(\mathbb{P}^3\). Since the moduli space of the cubic surfaces is also four-dimensional, there is a map \(\Psi : S \to \Lambda\) that corresponds to transformations of the first equation to the form of the second one. Of course, the map \(\Psi\) is multivalued. The author explicitly constructs a covering space \(\widetilde S \to S\) admitting a linear \(W(E_6)\)-action, where \(W(E_6)\) is the Weyl group of type \(E_6\). He then proves that the induced map \(\Phi : \widetilde S \to \Lambda \) is \(W(E_6)\)-equivariant. Hence \(\Phi\) defines a single-valued map to \(\Lambda\). The construction is based on observations due to \textit{M. Yoshida} and \textit{B. Hunt} concerning the realization of \(W(E_6)\) as a group of birational transformations of the configuration space of 6 points in \(\mathbb{P}^2\) as well as the standard affine space \(\mathbb{C}^4\). Although the essential part of the proof has computational character and is partially performed with the help of a computer the author writes out all resulting formulas in a highly compact form. This enables him to give a clear interpretation of several earlier results by \textit{I. Naruki} [Proc. Lond. Math. Soc., III. Ser. 45, 1-30 (1982; Zbl 0508.14005)] and \textit{T. Terada} [J. Math. Soc. Japan 35, 451-475 (1983; Zbl 0506.33001)], and some others.
Appell- Lauricella hypergeometric function
scientific article; zbMATH DE number 711444