Notes on Geometry and Arithmetic by Daniel Coray

Author:Daniel Coray
Language: eng
Format: epub, pdf
ISBN: 9783030437817
Publisher: Springer International Publishing

hence also:

Now, if we set

(7.4.2)

in Equation I, we find , which shows that the intersection of these two planes is indeed a line lying on Y , provided that λμ 2 = 1. On combining the relations we have obtained for this line L μ, we see that it is given in Plücker coordinates by

(7.4.3)

This is the parametric representation of a twisted cubic Γ, drawn on the Klein quadric. It has a point at infinity (corresponding to λ = 0), namely , associated with the line .

The curve Γ is contained in the 3-dimensional space defined by p 3 = p 6 = 0 and, in that space, it lies on the smooth quadric p 1 p 4 + p 2 p 5 = 0 and also on the quadratic cones and (on this subject, see Exercise 4.6). We may note that the vertexes of these cones are just the points associated with L ∞ and respectively. □

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.