\(W(E_6)\)-action on the configuration space of six lines on the real projective plane.

*(English)*Zbl 0932.14027From 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 \(\varphi:{\mathcal C}\to\overline X\).

(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 |