# Rational normal scroll

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{{Short description|Algebraic geometry}}
{{Multiple issues|{{one source|date=November 2021}}{{no footnotes|date=November 2021}}}}

In mathematics, a '''rational normal scroll''' is a [ruled surface](/source/ruled_surface) of degree ''n'' in [projective space](/source/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](/source/projective_normality) (not [normal scheme](/source/normal_scheme)s). 

A non-degenerate irreducible surface of degree ''m''&nbsp;–&nbsp;1 in '''P'''<sup>''m''</sup> is either a rational normal scroll or the [Veronese surface](/source/Veronese_surface).

==Construction==

In projective space of dimension ''m''&nbsp;+&nbsp;''n''&nbsp;+&nbsp;1 choose two complementary linear subspaces of dimensions ''m''&nbsp;>&nbsp;0 and ''n''&nbsp;>&nbsp;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''&nbsp;<&nbsp;''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==

*{{Citation | last1=Griffiths | first1=Phillip | author1-link=Phillip Griffiths | last2=Harris | first2=Joseph | author2-link=Joe Harris (mathematician) | title=Principles of algebraic geometry | publisher=[John Wiley & Sons](/source/John_Wiley_%26_Sons) | location=New York | series=Wiley Classics Library | isbn=978-0-471-05059-9 |mr=1288523 | year=1994}}

Category:Algebraic geometry

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