On maps taking lines to plane curves

A planarization is a mapping $f$ of an open subset $U$ of the real projective plane into the real projective $n$-space, such that $f(Lcap U)$ is a subset of a hyperplane, for every line $L$. Studying planarizations is closely related to studying maps taking lines to curves of certain linear systems; a classical result of this type is the Moebius–von Staudt theorem about maps taking lines to lines, sometimes called the Fundamental Theorem of Projective Geometry. We assume that the planarizations are sufficiently smooth, i.e., sufficiently many times differentiable. We give a complete description of all planarizations in case $n=3$ up to the following equivalence relation: two planarizations are equivalent if they coincide on a nonempty open set, after a projective transformation of the source space and a projective transformation of the target space. Apart from trivial cases, there are $16$ equivalence classes, among which $6$ classes of cubic rational maps (all remaining nontrivial classes are represented by quadratic rational maps).
(based on a joint project with V. Petruschenko)