Rational curves on the Grassmannian \(G_{4,2}\)
Let \(G_{4,2}\) denote the Grassmannian of 2-planes in \({\mathbb{C}}^ 4\), or, equivalently, of lines in \({\mathbb{P}}^ 3\). Then \(G_{4,2}\) can be identified, by means of Plücker coordinates, with a nondegenerate quadric Q in \({\mathbb{P}}^ 5\). On an arbitrary hyperboloid in \({\mathbb{P}}^ 3\) there lie two families of rectilinear generators. Let \(\gamma\) be a rational curve in \(G_{4,2}\). Then the following conditions are equivalent: (1) The points of \(\gamma\) correspond to the generators of one family of a certain hyperboloid in \({\mathbb{P}}^ 3.\) (2) Under the identification \(G_{4,2}=Q\), \(\gamma\) coincides with a nondegenerate curve of degree two lying on Q. (3) Under the identification \(G_{4,2}=Q\), \(\gamma\) is the intersection of Q with a two-dimensional nontangent plane.
scientific article; zbMATH DE number 4148297
scientific article