# Rational normal scroll

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 **P**^{m} is either a rational normal scroll or the Veronese surface.

## Construction

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

- 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