Rational normal curve
In mathematics, the rational normal curve is a smooth, rational curve of degree n in projective n-space It is a simple example of a projective variety; formally, it is the Veronese variety when the domain is the projective line. For n=2 it is the flat conic and for n=3 it is the twisted cubic. The term "normal" is an old term meaning that the linear system defining the embedding is complete (and has nothing to do with normal schemes). The intersection of the rational normal curve with an affine space is called the moment curve.
The rational normal curve may be given parametrically as the image of the map
which assigns to the homogeneous coordinates the value
In the affine coordinates of the chart the map is simply
where are the homogeneous coordinates on . The full set of these polynomials is not needed; it is sufficient to pick n of these to specify the curve.
Let be distinct points in . Then the polynomial
is a homogeneous polynomial of degree with distinct roots. The polynomials
are then a basis for the space of homogeneous polynomials of degree n. The map
or, equivalently, dividing by
is a rational normal curve. That this is a rational normal curve may be understood by noting that the monomials are just one possible basis for the space of degree-n homogeneous polynomials. In fact, any basis will do. This is just an application of the statement that any two projective varieties are projectively equivalent if they are congruent modulo the projective linear group (with K the field over which the projective space is defined).
This rational curve sends the zeros of G to each of the coordinate points of ; that is, all but one of the vanish for a zero of G. Conversely, any rational normal curve passing through the n+1 coordinate points may be written parametrically in this way.
The rational normal curve has an assortment of nice properties:
- Any points on are linearly independent, and span . This property distinguishes the rational normal curve from all other curves.
- Given points in in linear general position (that is, with no lying in a hyperplane), there is a unique rational normal curve passing through them. The curve may be explicitly specified using the parametric representation, by arranging of the points to lie on the coordinate axes, and then mapping the other two points to and .
- The tangent and secant lines of a rational normal curve are pairwise disjoint, except at points of the curve itself. This is a property shared by sufficiently positive embeddings of any projective variety.
Every irreducible non-degenerate curve of degree is a rational normal curve.
- Joe Harris, Algebraic Geometry, A First Course, (1992) Springer-Verlag, New York. ISBN 0-387-97716-3