Rational normal scroll

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title: "Rational normal scroll" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["algebraic-geometry"] description: "Algebraic geometry" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Rational_normal_scroll" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Algebraic geometry ::

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

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

References

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