Projective subvarieties of the Plücker variety and rational morphisms (Q2777563)

Let \(f : X \to P_n\) be a rational morphism from a variety \(X\subset P_n\) into the \(n\)-dimensional projective space \(P_n\) over an algebraically closed field. A set \(\mathfrak I\) of lines in \(P_n\) is said to realize the mapping \(f\) if, with the exception of a set of smaller dimension, every line of \(\mathfrak I\) can be written as a join \(x {\scriptstyle \vee} f(x)\). The Plücker mapping from the line set \({\mathcal L}_n\) into \(P_m\) is denoted by \(Q_n\). NEWLINENEWLINENEWLINEThe author makes a few remarks on relations between these notions. Typical is the following: The regular points \(x\) with \(f(x) \neq x\) yield a subvariety \(X_f\) in \(P_m\). -- If \(\overline{X}_f\) is the closure in the Zariski topology, then \(Q_n^{-1}(\overline{X}_f)\) realizes the mapping~\(f\). NEWLINENEWLINENEWLINEThe last 3 pages deal with the special case of birational morphisms of a projective line.