Justyna Szpond: Parameter spaces of some extremal line configurations and applications

I will be talking about two line configurations introduced by Böröczky. These configurations are extremal from the point of view of the Dirac–Motzkin Conjecture settled recently by Green and Tao.

They have appeared also recently in commutative algebra in connection with the containment problem for symbolic and ordinary powers of homogeneous ideals and in algebraic geometry in considerations revolving around the Bounded Negativity Conjecture. I will show that the parameter space of Böröczky configurations of 12 lines is a three dimensional rational variety. Hence there exists a three dimensional family of such rational configurations.

On the other hand the parameter space of Böröczky configurations of 15 lines is an elliptic curve with only finitely many rational points, all corresponding to degenerate configurations. Thus, somewhat surprisingly, it turns out that there are no rational Böröczky configurations of 15 lines.

The talk is based on joint work with Magdalena Lampa-Baczynska.