Rational normal curve
In mathematics, the rational normal curve is a smooth, rational curve of degree 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 it is the plane conic and for it is the twisted cubic. The term "normal" refers to projective normality, not normal schemes. The intersection of the rational normal curve with an affine space is called the moment curve.
Definition
The rational normal curve may be given parametrically as the image of the mapwhich assigns to the homogeneous coordinates 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, rational normal 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 of these to specify the curve.
Alternate parameterization
Let be distinct points in. Then the polynomialis a homogeneous polynomial of degree with distinct roots. The polynomials
are then a basis for the space of homogeneous polynomials of degree. 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 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 .
This rational curve sends the zeros of to each of the coordinate points of ; that is, all but one of the vanish for a zero of. Conversely, any rational normal curve passing through the coordinate points may be written parametrically in this way.
Properties
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, 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
- The curve is not a complete intersection, for. That is, it cannot be defined by only equations, that being the codimension of the curve in.
- The canonical mapping for a hyperelliptic curve has image a rational normal curve, and is 2-to-1.
- Every irreducible non-degenerate curve of degree is a rational normal curve.