The versal deformation of the \(E_ 6\)-singularity and a family of cubic surfaces
type: Item
instance of: scholarly article
title: The versal deformation of the \(E_ 6\)-singularity and a family of cubic surfaces
zbMATH Open document ID: 0830.14012
author: Jiro Sekiguchi
published in: Journal of the Mathematical Society of Japan
publication date: 1996-01-30T00:00:00Z
review text: 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 \(W(E_6)\) a…
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14B07
Mathematics Subject Classification ID: 33C65
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 32S25
zbMATH DE Number: 711444
zbMATH Keywords: cross ratio variety
zbMATH Keywords: rational double point
zbMATH Keywords: Weyl groups
zbMATH Keywords: Appell- Lauricella hypergeometric function
zbMATH Keywords: deformation of \(E_ 6\)-singularity
zbMATH Keywords: cubic surface
zbMATH Keywords: Cayley family
reviewed by: Aleksandr G. Aleksandrov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04620355
OpenAlex ID: W2137768692
DOI: 10.2969/JMSJ/04620355
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description: scientific article; zbMATH DE number 711444