Rational normal curve

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In mathematics, the rational normal curve is a smooth, rational curve $C$ of degree n in projective n-space $\mathbb{P}^n$ . It is a simple example of a projective variety. The twisted cubic is the special case of n=3.

1 Definition
2 Alternate parameterization
3 Properties
4 Reference

[edit] Definition

The rational normal curve may be given parametrically as the map

$\nu:\mathbb{P}^1\to\mathbb{P}^n$

which assigns to the homogeneous coordinate $[S : T]$ the value

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

In the affine coordinates of projective space, the map is simply

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

That is, it is the closure by a single point at infinity of the affine curve $(x,x^2,\dots,x^n)$ .

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

$F_{i,j}(X_0,\ldots,X_n) = X_iX_j - X_{i+1}X_{j-1}$

where $[X_0:\ldots:X_n]$ are the homogeneous coordinates on $\mathbb{P}^n$ . The full set of these polynomials is not needed; it is sufficient to pick n of these to specify the curve.

[edit] Alternate parameterization

Let $[a i : b i]$ be $n + 1$ distinct points in $\mathbb{P}^1$ . Then the polynomial

$G(S,T) = \Pi_{i=0}^n (a_iS -b_iT)$

is a homogeneous polynomial of degree $n + 1$ with distinct roots. The polynomials

$H_i(S,T) = \frac{G(S,T)} {(a_iS-b_iT)}$

are then a basis for the space of homogeneous polynomials of degree n. The map

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

or, equivalently, dividing by $G (S, T)$

$[S:T] \mapsto \left[\frac{1}{(a_0S-b_0T)} : \ldots : \frac{1}{(a_nS-b_nT)}\right]$

is a rational normal curve. That this is a rational normal curve may be understood by noting that the monomials $S^n,S^{n-1}T,S^{n-2}T^2,\ldots,T^n$ 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 $PGL n + 1 K$ (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 $\mathbb{P}^n$ ; that is, all but one of the $H i$ 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.

[edit] Properties

The rational normal curve has an assortment of curious properties:

Any $n + 1$ points on $C$ are linearly independent, and span $\mathbb{P}^n$ .
Given $n + 3$ points in $\mathbb{P}^n$ in general position (that is, with no $n + 1$ coplanar), there is a unique rational normal curve passing through them. The curve may be explicitly specified using the parametric representation, by arranging $n + 1$ of the points to lie on the coordinate axes, and then mapping the other two points to $[S : T] = [0:1]$ and $[S : T] = [1:0]$ .

There are a large number of quadrics that pass through the curve: it is easy to find some from relations like

t².t³ = t⁵.

But the curve is not a complete intersection, for n > 2. Therefore it is not defined by the number of equations equal to its codimension n − 1. The twisted cubic is therefore an easy example of a variety that is not a complete intersection; that it is not is plain from counting degrees, as 3 is a prime number, using the hypersurface Bézout theorem that degrees multiply when you intersect. Two quadrics in projective three-space that are in general position intersect in an elliptic curve. This means that two quadrics through a twisted cubic C cannot be in general position; in fact by counting degrees they must intersect in C and a line L.

The canonical mapping for a hyperelliptic curve has image a rational normal curve, and is 2-to-1.