Planarizations and maps taking lines to linear webs of conics (Q1951843)

Let \(U\subset{\mathbb R}P^2\) be an open subset; the author defines a planarization as a map \(F:U\to{\mathbb R}P^n\), for which there exists an open set of lines \(L\subset{\mathbb R}P^2\) such that \(U\cap\,L\not=\emptyset\), and the set \(F(U\cap\,L)\) lies in a projective hyperplane. For a sufficiently smooth planarization \(F:U\to{\mathbb R}P^3\) holds, possibly after the restriction to a smaller open set \(U'\), at least one of the following three possibilities: \(F|_{U'}\) (1) is a trivial planarization or (2) a co-trivial planarization or (3) a rational map of degree at most \(3\). As immediate corollary of the result above, the author derives a theorem of Khovanskii which deals with maps from \(U\) to the \(2\)-sphere. Finally, the author considers a sufficiently smooth map \(f:U\to{\mathbb R}P^2\) which takes lines to conics of a web; possibly after the restriction to a smaller open set \(U'\), one of the following four possibilities holds: (1) \(f(U')\) is a subset of a conic from the web or (2) \(f|_{U'}\) is a local inverse of a quadratic rational map or (3) \(f|_{U'}\) is a quadratic rational map or (4) \(f|_{U'}\) is a local branch of a multi-valued map \(\Phi^{-1}\circ\,F\), where \(\Phi\) and \(F\) are quadratic rational maps to the same irreducible quadric of \({\mathbb R}P^3\).