Dynamical systems on some elliptic modular surfaces via operators on line arrangements

This paper further studies the matroid realization space of a specific deformation of the regular n-gon with its lines of symmetry. Recently, we obtained that these particular realization spaces are birational to the elliptic modular surfaces ₁ (n) over the modular curve X₁ (n). Here, we focus on the peculiar cases when n=7, 8 in more detail. We obtain concrete quartic surfaces in P³ equipped with a dominant rational self-map stemming from an operator on line arrangements, which yields K3 surfaces with a dynamical system that is semi-conjugated to the plane.