Abstract. The purpose of this note is to give defined over the real numbers counterexamples to a question relevant in the commutative algebra, concerning a containment relation between algebraic and symbolic powers of homogeneous ideal.

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. We show that the parameter space of what we call B12 configurations is a three dimensional rational variety. As a consequence we derive the existence of a three dimensional family of rational B12 configurations. On the other hand the parameter space of B15 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 B15 configurations.

The purpose of this note is to study configurations of lines in projective planes over arbitrary fields having the maximal number of intersection points where three lines meet. We give precise conditions on ground fields F over which such extremal configurations exist. We show that there does not exist a field admitting a configuration of 11 lines with 17 triple points, even though such a configuration is allowed combinatorially. Finally, we present an infinite series of configurations which have a high number of triple intersection points.