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| In [[mathematics]], a '''twisted cubic''' is a smooth, [[rational curve]] ''C'' of [[Degree of an algebraic variety|degree]] three in [[projective space|projective 3-space]] '''P'''<sup>3</sup>. It is a fundamental example of a [[skew curve]]. It is essentially unique, up to [[projective transformation]] (''the'' twisted cubic, therefore). It is generally considered to be the simplest example of a [[projective variety]] that isn't linear or a [[hypersurface]], and is given as such in most textbooks on [[algebraic geometry]]. It is the three-dimensional case of the [[rational normal curve]], and is the image of a [[Veronese map]] of degree three on the projective line.
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| ==Definition==
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| [[File:Twisted cubic curve.png|250px|right]]
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| It is most easily given [[parametrically]] as the image of the map
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| :<math>\nu:\mathbf{P}^1\to\mathbf{P}^3</math>
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| which assigns to the [[homogeneous coordinate]] <math>[S:T]</math> the value
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| :<math>\nu:[S:T] \mapsto [S^3:S^2T:ST^2:T^3].</math>
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| In one [[coordinate patch]] of projective space, the map is simply the [[moment curve]]
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| :<math>\nu:x \mapsto (x,x^2,x^3)</math>
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| That is, it is the closure by a single [[point at infinity]] of the [[affine curve]] <math>(x,x^2,x^3)</math>.
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| Equivalently, it is a [[projective variety]], defined as the zero locus of three smooth [[quadric]]s. Given the homogeneous coordinates [''X:Y:Z:W''] on '''P'''<sup>3</sup>, it is the zero locus of the three [[homogeneous polynomial]]s
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| :<math>F_0 = XZ - Y^2</math> | |
| :<math>F_1 = YW - Z^2</math>
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| :<math>F_2 = XW - YZ.</math>
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| It may be checked that these three [[quadratic form]]s vanish identically when using the explicit parameterization above; that is, substituting ''x''<sup>3</sup> for ''X'', and so on.
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| In fact, the [[homogeneous ideal]] of the twisted cubic ''C'' is generated by three [[algebraic form]]s of degree two on '''P'''<sup>3</sup>. The generators of the ideal are
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| :<math>\{ XZ - Y^2 , YW - Z^2 , XW - YZ \}.</math>
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| ==Properties==
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| The twisted cubic has an assortment of elementary properties:
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| * It is the set-theoretic complete intersection of ''XZ''-''Y''<sup>2</sup> and <math>Z(YW-Z^2)-W(XW-YZ)</math>, but not a scheme-theoretic or ideal-theoretic complete intersection (the resulting ideal is not radical, since <math>(YW-Z^2)^2</math> is in it, but <math>YW-Z^2</math> is not).
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| * Any four points on ''C'' span '''P'''<sup>3</sup>.
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| * Given six points in '''P'''<sup>3</sup> with no four coplanar, there is a unique twisted cubic passing through them.
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| * The union of the [[tangent]] and [[secant line|secant]] lines, the [[secant variety]], of a twisted cubic ''C'' fill up '''P'''<sup>3</sup> and the lines are pairwise disjoint, except at points of the curve itself. In fact, the union of the [[tangent]] and [[secant line|secant]] lines of any non-planar smooth [[algebraic curve]] is three-dimensional. Further, any smooth [[algebraic variety]] with the property that every length four [[subscheme]] spans '''P'''<sup>3</sup> has the property that the [[tangent]] and [[secant line|secant]] lines are pairwise disjoint, except at points of the variety itself.
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| * The projection of ''C'' onto a plane from a point on a tangent line of ''C'' yields a [[cuspidal cubic]].
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| * The projection from a point on a secant line of ''C'' yields a [[nodal cubic]].
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| * The projection from a point on ''C'' yields a [[conic section]].
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| ==References==
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| *{{Citation |first=Joe |last=Harris |title=Algebraic Geometry, A First Course |year=1992 |publisher=Springer-Verlag |location=New York |isbn=0-387-97716-3 }}.
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| [[Category:Algebraic curves]]
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