The following problem is what motivated my previous MO question.

It is easily seen that for any given 0-1 matrix $M$, one can always find
a set $\mathcal P$ of points, and a set $\mathcal C$ of simple curves in
the plane, so that their incidence matrix is exactly the matrix $M$.
Suppose, however, that any pair of curves from $\mathcal C$ is now
allowed to intersect in at most one point (be it a point of $\mathcal P$
or any other point), and let's say that the matrix $M$ is
*realizable* if such $\mathcal P$ and $\mathcal C$ can be found.
Clearly, a necessary condition for this is that the scalar product of any
two rows of $M$ be at most $1$, but this condition is insufficient: say,
for $q$ large enough, by the Szemeredi-Trotter theorem, the point-line
incidence matrix of the finite projective plane $PG(2,q)$ has two many
incidences to be realizable. What are other reasonable necessary /
sufficient conditions for $M$ to be realizable? What are "small" examples
of non-realizable 0-1 matrices?

### Added June 03, 2013

Here is a very specific question along these lines. Everything I presently can say on the problem above is essentially symmetric in $\mathcal P$ and $\mathcal C$. This suggests that, perhaps, there is some duality between points and curves involved. Accordingly, I wonder whether, by any chance, it can be true that $M$ is realizable if and only if its transpose $M^t$ is realizable?