In mathematics, a rational normal scroll is a

ruled surface In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the ...

of degree ''n'' in

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

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 In algebraic geometry, the homogeneous coordinate ring ''R'' of an algebraic variety ''V'' given as a subvariety of projective space of a given dimension ''N'' is by definition the quotient ring :''R'' = ''K'' 'X''0, ''X''1, ''X''2, ..., ''X'N'' ...

(not

normal scheme In algebraic geometry, an algebraic variety or scheme ''X'' is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety ''X'' (understood to be irreducible) is normal if and ...

s). A non-degenerate irreducible surface of degree ''m'' – 1 in P^''m'' is either a rational normal scroll or the

Veronese surface In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complete linear system of conics. It is named after Giu ...

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

*{{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 John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, in p ...

, location=New York , series=Wiley Classics Library , isbn=978-0-471-05059-9 , mr=1288523 , year=1994 Algebraic geometry