Point sets in general position that determine lines with a small piercing set

Let $P$ be a set of $n$ points in general position (no three on a line) in the plane. Assume $R$ is another set of $n$ points disjoint from $P$ such that every line through two points in $P$ passes through a point in $R$. It is conjectured that in such a case $P$ is contained in a cubic curve in the plane. In a joint work with Chaya Keller we prove this conjecture under additional assumption that the point in $R$ collinear with two points $a$ and $b$ in $P$ is not contained in the straight line segment delimited by $a$ and $b$. This already generalizes a result of Jamison from 1978 about point sets that determine minimum number of distinct directions. We will discuss related results and open problems.