, 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
. The twisted cubic
is the special case of n
The rational normal curve may be given parametrically
as the image of the map
which assigns to the homogeneous coordinate the value
In the affine coordinates of the chart the map is simply
That is, the rational normal curve is the closure by a single point at infinity of the affine curve .
Equivalently, normal rational curve may be understood to be a projective variety, defined as the common zero locus of the homogeneous polynomials
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.
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.
There are independent quadrics that generate the ideal of the curve.
The curve is not a complete intersection, for . This means it is not defined by the number of equations equal to its codimension .
The canonical mapping for a hyperelliptic curve has image a rational normal curve, and is 2-to-1.
- Joe Harris, Algebraic Geometry, A First Course, (1992) Springer-Verlag, New York. ISBN 0-387-97716-3