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$$W(E_6)$$-action on the configuration space of six lines on the real projective plane. (English) Zbl 0932.14027
From the introduction: The main objective of this paper is the study of the configuration space $$X$$ of six lines in general position (no three lines meet at a point) on the projective plane. The space $$X$$ can be naturally regarded as a Zariski open subset of a four-dimensional affine space and admits the action of the symmetric group $$S_6$$ induced from the permutations of the six lines. Let $$Q$$ be the subset of $$X$$ consisting of six lines admitting a conic tangent to all these lines; $$Q$$ is a hypersurface of $$X$$. On the space $$X-Q$$ the Weyl group $$W(E_6)$$ of type $$E_6$$ acts, extending the $$S_6$$-action. The present study is focused on this $$W(E_6)$$-action. Noting that the spaces $$X$$, $$X-Q$$ and this group action are defined over the real numbers, in this paper, we study these spaces over the real number field only. Thus the space $$X-Q$$ is divided into several connected components. We show the following.
(i) The space $$X-Q$$ has 432 connected components on which the group $$W(E_6)$$ acts transitively.
(ii) For each connected component of $$X-Q$$, its isotropy subgroup in $$W(E_6)$$ is isomorphic to the symmetric group $$S_5$$.
(iii) Each connected component of $$X-Q$$ can be coded by a pentagonal set consisting of ten roots of the root system $$\Delta$$ of type $$E_6$$.
(iv) To each pentagonal set, we associate a pent-diagram similar to extended Dynkin diagrams for root systems. (There is a two-to-one map of the set of vertices of the dodecahedron to a pentagonal set.)
We make a dictionary between six lines on a real projective plane and the pentagonal sets of the root system $$\Delta$$. We make a detailed study on Naruki’s cross ratio variety $${\mathcal C}$$ which is a smooth compactification of $$X-Q$$ admitting a biregular action of $$W(E_6)$$. We show the following.
(v) Each connected component of $$X-Q$$ is bounded by fifteen hypersurfaces in $${\mathcal C}$$. These hypersurfaces can be described in terms of the roots in $$\Delta$$.
There is another compactification $$\overline X$$ of $$X$$ studied by K. Matsumoto, T. Sasaki and M. Yoshida [Int. J. Math. 3, No. 1, 1-164 (1992; Zbl 0763.32016)] which is in some sense the smallest compactification among those admitting $$S_6$$-actions. We show the following:
(vi) There is an $$S_6$$-equivariant birational morphism $$\varphi:{\mathcal C}\to\overline X$$.

##### MSC:
 14N20 Configurations and arrangements of linear subspaces 14L30 Group actions on varieties or schemes (quotients) 14P99 Real algebraic and real-analytic geometry 14R20 Group actions on affine varieties
PENT
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