Rational curves on the Grassmannian \(G_{4,2}\)
type: Item
instance of: scholarly article
title: Rational curves on the Grassmannian \(G_{4,2}\)
zbMATH Open document ID: 0701.14048
DOI: 10.1007/BF01057189
author: Ya. Yu. Gajdis
published in: Ukrainian Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 4148297
zbMATH Keywords: Grassmannian
zbMATH Keywords: hyperboloid
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
cites work: Q3940444
cites work: Q5814388
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description: scientific article; zbMATH DE number 4148297