Geometry of Arrangements

Many questions on arrangements of lines in the projective plane remain open. For instance, the complete classification of complex arrangements without double points is still unknown, as is the full list of real simplicial arrangements, where the regions formed by the lines are exclusively triangles.

In the hop to generate new line arrangements with interesting properties from known ones, we introduce operators acting on the set of plane line arrangements.

In this talk, I will present examples of realization spaces of line arrangements that are preserved by certain such operators. In particular, we will see that the elliptic surface (\Xi_1(n)) over the modular elliptic curve (X_1(n))—which parametrizes elliptic curves with a torsion point of order (n > 3)—can also be interpreted as (the compactification of) the realization space of certain line arrangements.

We will explore the existence of an operator acting on these arrangements and, consequently, on the elliptic surface (\Xi_1(n)), and describe its action on the surface.

Work in collaboration with Lukas Kühne.