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| In [[mathematics]], the '''rational normal curve''' is a smooth, [[rational curve]] <math>C</math> of [[Degree of an algebraic variety|degree]] ''n'' in [[projective space|projective n-space]] <math>\mathbb{P}^n.</math> It is a simple example of a [[projective variety]]; formally, it is the [[Veronese variety]] when the domain is the projective line. For ''n''=2 it is the [[flat conic]] <math>Z_0 Z_2 = Z_1^2,</math> and for ''n''=3 it is the [[twisted cubic]]. The term "normal" is an old term meaning that the linear system defining the embedding is complete (and has nothing to do with [[normal scheme]]s). The intersection of the rational normal curve with an affine space is called the [[moment curve]].
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| ==Definition==
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| The rational normal curve may be given [[parametrically]] as the image of the map
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| :<math>\nu:\mathbb{P}^1\to\mathbb{P}^n</math> | |
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| which assigns to the [[homogeneous coordinates]] <math>[S:T]</math> the value
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| :<math>\nu:[S:T] \mapsto [S^n:S^{n-1}T:S^{n-2}T^2:\ldots:T^n].</math>
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| In the [[affine coordinates]] of the chart <math> x_0\neq0 </math> the map is simply
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| :<math>\nu:x \mapsto (x,x^2, \ldots ,x^n).</math>
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| That is, the rational normal curve is the closure by a single [[point at infinity]] of the [[affine curve]] <math>(x,x^2,\dots,x^n)</math>.
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| Equivalently, rational normal curve may be understood to be a [[projective variety]], defined as the common zero locus of the [[homogeneous polynomial]]s
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| :<math>F_{i,j}(X_0,\ldots,X_n) = X_iX_j - X_{i+1}X_{j-1}</math>
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| where <math>[X_0:\ldots:X_n]</math> are the [[homogeneous coordinate]]s on <math>\mathbb{P}^n</math>. The full set of these polynomials is not needed; it is sufficient to pick ''n'' of these to specify the curve.
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| ==Alternate parameterization==
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| Let <math>[a_i:b_i]</math> be <math>n+1</math> distinct points in <math>\mathbb{P}^1</math>. Then the polynomial
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| :<math>G(S,T) = \Pi_{i=0}^n (a_iS -b_iT)</math>
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| is a [[homogeneous polynomial]] of degree <math>n+1</math> with distinct roots. The polynomials
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| :<math>H_i(S,T) = \frac{G(S,T)} {(a_iS-b_iT)}</math>
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| are then a [[Basis (linear algebra)|basis]] for the space of homogeneous polynomials of degree ''n''. The map
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| :<math>[S:T] \mapsto [H_0(S,T) : H_1(S,T) : \ldots : H_n (S,T) ]</math>
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| or, equivalently, dividing by <math>G(S,T)</math>
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| :<math>[S:T] \mapsto \left[\frac{1}{(a_0S-b_0T)} : \ldots : \frac{1}{(a_nS-b_nT)}\right] </math>
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| is a rational normal curve. That this is a rational normal curve may be understood by noting that the [[monomial]]s <math>S^n,S^{n-1}T,S^{n-2}T^2,\ldots,T^n</math> are just one possible [[base (topology)|basis]] for the space of degree-''n'' homogeneous polynomials. In fact, any [[Basis (linear algebra)|basis]] will do. This is just an application of the statement that any two projective varieties are projectively equivalent if they are [[congruence relation|congruent]] modulo the [[projective linear group]] <math>\rm{PGL}_{n+1} K</math> (with ''K'' the [[field (mathematics)|field]] over which the projective space is defined). | |
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| This rational curve sends the zeros of ''G'' to each of the coordinate points of <math>\mathbb{P}^n</math>; that is, all but one of the <math>H_i</math> 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.
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| ==Properties== | |
| The rational normal curve has an assortment of nice properties:
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| * Any <math>n+1</math> points on <math>C</math> are linearly independent, and span <math>\mathbb{P}^n</math>. This property distinguishes the rational normal curve from all other curves.
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| * Given <math>n+3</math> points in <math>\mathbb{P}^n</math> in linear [[general position]] (that is, with no <math>n+1</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 <math>n+1</math> of the points to lie on the coordinate axes, and then mapping the other two points to <math>[S:T]=[0:1]</math> and <math>[S:T]=[1:0]</math>.
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| * 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.
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| There are <math>\binom{n+2}{2}-2n-1</math> independent [[quadric]]s that generate the [[ideal]] of the curve.
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| The curve is not a [[complete intersection]], for <math>n>2</math>. This means it is not defined by the number of equations equal to its [[codimension]] <math>n-1</math>.
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| The [[canonical mapping]] for a [[hyperelliptic curve]] has image a rational normal curve, and is 2-to-1.
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| Every irreducible non-degenerate curve <math>C\subset \mathbb{P}^n</math> of degree <math>n</math> is a rational normal curve.
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| ==See also==
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| *[[Rational normal surface]]
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| ==References==
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| * Joe Harris, ''Algebraic Geometry, A First Course'', (1992) Springer-Verlag, New York. ISBN 0-387-97716-3
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| [[Category:Algebraic curves]]
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| [[Category:Birational geometry]]
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