  Let $P$ be a set of $n$ points in general position in the plane. Let $R$ be a
set of points disjoint from $P$ such that for every $x,y \in P$ the line
through $x$ and $y$ contains a point in $R$. We show that if $|R| <
\frac{3}{2}n$ and $P \cup R$ is contained in a cubic curve $c$ in the plane,
then $P$ has a special property with respect to the natural group action on
$c$. That is, $P$ is contained in a coset of a subgroup $H$ of $c$ of
cardinality at most $|R|$.
  We use the same approach to show a similar result in the case where each of
$B$ and $G$ is a set of $n$ points in general position in the plane and every
line through a point in $B$ and a point in $G$ passes through a point in $R$.
This provides a partial answer to a problem of Karasev.
  The bound $|R| < \frac{3}{2}n$ is best possible at least for part of our
results. Our extremal constructions provide a counterexample to an old
conjecture attributed to Jamison about point sets that determine few
directions. Jamison conjectured that if $P$ is a set of $n$ points in general
position in the plane that determines at most $2n-c$ distinct directions, then
$P$ is contained in an affine image of the set of vertices of a regular
$m$-gon. This conjecture of Jamison is strongly related to our results in the
case the cubic curve $c$ is reducible and our results can be used to prove
Jamison's conjecture at least when $m-n$ is in the order of magnitude of
$O(\sqrt{n})$.
