From Pappus Theorem to parameter spaces of some extremal line point configurations and applications

Abstract

In the present work we study parameter spaces of two line point configurations introduced by Böröczky. These configurations are extremal from the point of view of the Dirac–Motzkin Conjecture settled recently by Green and Tao (Discrete Comput Geom 50:409–468, 2013). They have appeared also recently in commutative algebra in connection with the containment problem for symbolic and ordinary powers of homogeneous ideals (Dumnicki et al. in J Algebra 393:24–29, 2013) and in algebraic geometry in considerations revolving around the Bounded Negativity Conjecture (Bauer et al. in Duke Math J 162:1877–1894, 2013). We show that the parameter space of what we call {mathbb {B}}12 configurations is a three dimensional rational variety. As a consequence we derive the existence of a three dimensional family of rational {mathbb {B}}12 configurations. On the other hand the parameter space of {mathbb {B}}15 configurations is shown to be an elliptic curve with only finitely many rational points, all corresponding to degenerate configurations. Thus, somewhat surprisingly, we conclude that there are no rational {mathbb {B}}15 configurations.