- Rational normal curve
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 aprojective variety . Thetwisted cubic is the special case of "n"=3.**Definition**The rational normal curve may be given

parametrically as the image of the map:$u:mathbb\{P\}^1\; omathbb\{P\}^n$

which assigns to the

homogeneous coordinate $[S:T]$ the value:$u:\; [S:T]\; mapsto\; [S^n:S^\{n-1\}T:S^\{n-2\}T^2:ldots:T^n]$

In the

affine coordinates of the chart $x\_0\; eq0$ the map is simply:$u:x\; mapsto\; (x,x^2,\; ldots\; ,x^n)$

That is, the rational normal curve is the closure by a single

point at infinity of theaffine 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 thehomogeneous polynomial s:$F\_\{i,j\}(X\_0,ldots,X\_n)\; =\; X\_iX\_j\; -\; X\_\{i+1\}X\_\{j-1\}$

where $[X\_0:ldots:X\_n]$ are the

homogeneous coordinate s 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.**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 $G(S,T)$

:$[S:T]\; mapsto\; left\; [frac\{1\}\{(a\_0S-b\_0T)\}\; :\; ldots\; :\; frac\{1\}\{(a\_nS-b\_nT)\}\; ight]$

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

monomial s $S^n,S^\{n-1\}T,S^\{n-2\}T^2,ldots,T^n$ 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 arecongruent modulo theprojective linear group $m\{PGL\}\_\{n+1\}\; K$ (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.

**Properties**The rational normal curve has an assortment of nice properties:

* Any $n+1$ points on $C$ are linearly independent, and span $mathbb\{P\}^n$. This property distinguishes the rational normal curve from all other curves.

* Given $n+3$ points in $mathbb\{P\}^n$ in lineargeneral position (that is, with no $n+1$ lying in ahyperplane ), 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 $[S:T]\; =\; [1:0]$.

* 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 $inom\{n+2\}\{2\}-2n-1$ independent

quadric s that generate theideal of the curve.The curve is not a

complete intersection , for $n>2$. This means it is not defined by the number of equations equal to itscodimension $n-1$.The

canonical mapping for ahyperelliptic curve has image a rational normal curve, and is 2-to-1.**References*** Joe Harris, "Algebraic Geometry, A First Course", (1992) Springer-Verlag, New York. ISBN 0-387-97716-3

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