Rational normal scroll
From Wikipedia the free encyclopedia
This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages)
|
In mathematics, a rational normal scroll is a ruled surface of degree n in projective space of dimension n + 1. Here "rational" means birational to projective space, "scroll" is an old term for ruled surface, and "normal" refers to projective normality (not normal schemes).
A non-degenerate irreducible surface of degree m – 1 in Pm is either a rational normal scroll or the Veronese surface.
Construction[edit]
In projective space of dimension m + n + 1 choose two complementary linear subspaces of dimensions m > 0 and n > 0. Choose rational normal curves in these two linear subspaces, and choose an isomorphism φ between them. Then the rational normal surface consists of all lines joining the points x and φ(x). In the degenerate case when one of m or n is 0, the rational normal scroll becomes a cone over a rational normal curve. If m < n then the rational normal curve of degree m is uniquely determined by the rational normal scroll and is called the directrix of the scroll.
References[edit]
- Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-05059-9, MR 1288523