Configurations of seven lines on the real projective plane and the root system of type \(E_7\)
The configuration of seven lines on the projective plane are closely related with the classical topic on the twenty-eight bitangents of plane quartic curves. As it is well-known, the Weyl group of type \(E_7\) is regarded as the group of incidence-preserving automorphism of the twenty-eight bitangents. The author studies the case where the seven lines are defined over the real number field. Let \(l_1,l_2,\dots,l_7\) be mutually different seven lines on \(\mathbb P^2(\mathbb R).\) Cummings and White showed that there are eleven non-equivalent classes of systems of seven lines with condition: No three of \(l_1,l_2,\dots,l_7\) intersect at a point. The purpose of this paper is to give an interpretation of the classification of Cummings and White in terms of the root system of type \(E_7.\) To accomplish this, the following condition on these lines is added: There is no conic tangent to any six of \(l_1,l_2,\dots,l_7.\) Moreover, the notion of tetrahedral sets is necessary which consists of ten roots modulo signs in the root system of type \(E_7\) and which plays an important role in the study.
twenty-eight bitangents of plane quartic curves
scientific article; zbMATH DE number 1398935