Double-sixers of hexagons, \(A_ 6\) and \(\text{PSL}_ 3(K)\)
The main aim of this paper is to obtain a nice geometric configuration in \(\text{PG}(2,K)\) whose stabilizer is \(A_6\). The point set of 6-arcs of \(\text{PG}(2,K)\) we call the vertices of hexagon, and 15 lines which pass through pairs of vertices we call its edges. Any non-vertex point of a hexagon through which pass three of its edges is called a Brianchon point of that hexagon. A hexagon with exactly 10 Brianchon points is called a Clebsch hexagon. Let \(H\) be a Clebsch hexagon. It lies in a single orbit under \(\text{PGL}_3(K)\). There are 15 triads of edges of \(H\) containing all six vertices of \(H\). The Brianchon points account for 10 of these triads. The others provide the sides of the five triangles of \(H\). There is a unique orthogonal polarity with respect to which the five triangles of \(H\) are self-polar. For our \(K\) this polarity corresponds to a genuine conic. We call this the conic of \(H\). We say that two distinct Clebsch hexagons are apolar if they share a triangle and if the conic of each contains those four Brianchon points of the other that are not on the sides of the shared triangle. There are sets of six Clebsch hexagons, any two apolar; these are sixers. Any sixer has a unique complement with the same set of 36 vertices; each hexagon of a sixer shares one vertex with each hexagon of the complementary sixer. We call a pair of complementary sixers a double-sixer. The stabilizer in \(\text{PGL}_3(K)\) of double-sixer \(\mathcal D\) is \(A_6\), except when \(K\) has characteristic 3 or 5 when it has order 720.
scientific article; zbMATH DE number 977866
scientific article