Rational normal curve

In mathematics, the rational normal curve is a smooth, rational curve Template:Mvar of degree Template:Mvar in projective n-space Template:Math. It is a simple example of a projective variety; formally, it is the Veronese variety when the domain is the projective line. For Template:Math it is the plane conic Template:Math and for Template:Math 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 map

$\nu :{𝐏}^1\to {𝐏}^n$

which assigns to the homogeneous coordinates Template:Math the value

$\nu :[S:T]\mapsto [S^n:S^{n-1}T:S^{n-2}{T}^2:\cdots :T^n].$

In the affine coordinates of the chart Template:Math the map is simply

$\nu :x\mapsto (x,x^2,\dots ,x^n).$

That is, the rational normal curve is the closure by a single point at infinity of the affine curve

$(x,x^2,\dots ,x^n).$

Equivalently, rational normal curve may be understood to be a projective variety, defined as the common zero locus of the homogeneous polynomials

$F_{i,j}(X_0,\dots ,X_n)=X_i{X}_j-X_{i+1}{X}_{j-1}$

where $[X_0:\cdots :X_n]$ are the homogeneous coordinates on Template:Math. The full set of these polynomials is not needed; it is sufficient to pick Template:Mvar of these to specify the curve.

Alternate parameterization

Let $[a_i:b_i]$ be Template:Math distinct points in Template:Math. Then the polynomial

$G(S,T)=\prod _{i=0}^n(a_{i}S-b_{i}T)$

is a homogeneous polynomial of degree Template:Math with distinct roots. The polynomials

$H_i(S,T)=\frac{G(S,T)}{(a_{i}S-b_{i}T)}$

are then a basis for the space of homogeneous polynomials of degree Template:Mvar. The map

$[S:T]\mapsto [H_0(S,T):H_1(S,T):\cdots :H_n(S,T)]$

or, equivalently, dividing by Template:Math

$[S:T]\mapsto [\frac{1}{(a_{0}S-b_{0}T)}:\cdots :\frac{1}{(a_{n}S-b_{n}T)}]$

is a rational normal curve. That this is a rational normal curve may be understood by noting that the monomials

$S^n,S^{n-1}T,S^{n-2}{T}^2,\cdots ,T^n,$

are just one possible basis for the space of degree Template:Mvar 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 Template:Math (with Template:Mvar the field over which the projective space is defined).

This rational curve sends the zeros of Template:Mvar to each of the coordinate points of Template:Math; that is, all but one of the Template:Math vanish for a zero of Template:Mvar. Conversely, any rational normal curve passing through the Template:Math coordinate points may be written parametrically in this way.

Properties

The rational normal curve has an assortment of nice properties:

• Any Template:Math points on Template:Mvar are linearly independent, and span Template:Math. This property distinguishes the rational normal curve from all other curves.
• Given Template:Math points in Template:Math in linear general position (that is, with no Template:Math 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 Template:Math of the points to lie on the coordinate axes, and then mapping the other two points to Template:Math and Template:Math.
• 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

$(\genfrac{}{}{0ex}{}{n+2}{2})-2n-1$

independent quadrics that generate the ideal of the curve.

• The curve is not a complete intersection, for Template:Math. That is, it cannot be defined (as a subscheme of projective space) by only Template:Math equations, that being the codimension of the curve in ${𝐏}^n$.
• The canonical mapping for a hyperelliptic curve has image a rational normal curve, and is 2-to-1.
• Every irreducible non-degenerate curve Template:Math of degree Template:Mvar is a rational normal curve.

See also

• Rational normal scroll

References

• Joe Harris, Algebraic Geometry, A First Course, (1992) Springer-Verlag, New York. Template:Isbn