On the dynamics of the line operator $\Lambda_{\{2\},\{3\}}$ on some arrangements of six lines

Abstract

The operator $\Lambda_{\{2\},\{3\}}$ acting on line arrangements is defined by associating to a line arrangement \mathcal{A}, the line arrangement which is the union of the lines containing exactly three points among the double points of \mathcal{A}. We say that six lines not tangent to a conic form an unassuming arrangement if the singularities of their union are only double points, but the dual line arrangement has six triple points, six 5-points and 27 double points. The moduli space of unassuming arrangements is the union of a point and a line. The image by the operator $\Lambda_{\{2\},\{3\}}$ of an unassuming arrangement is again an unassuming arrangement. We study the dynamics of the operator $\Lambda_{\{2\},\{3\}}$ on these arrangements and we obtain that the periodic arrangements are related to the Ceva arrangements of lines.