Editing Twisted cubic

{{No footnotes|date=February 2022}}
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). In [[algebraic geometry]], the twisted cubic is a simple example of a [[projective variety]] that is not linear or a [[hypersurface]], in fact not a [[complete intersection]]. It is the three-dimensional case of the [[rational normal curve]], and is the [[Image (mathematics)|image]] of a [[Veronese surface#Veronese map|Veronese map]] of degree three on the [[projective line]].

==Definition==
[[File:Twisted cubic curve.png|250px|right]]

The twisted cubic is most easily given [[parametrically]] as the image of the map

:<math>\nu:\mathbf{P}^1\to\mathbf{P}^3</math>

which assigns to the [[homogeneous coordinate]] <math>[S:T]</math> the value

:<math>\nu:[S:T] \mapsto [S^3:S^2T:ST^2:T^3].</math>

In one [[coordinate patch]] of projective space, the map is simply the [[moment curve]]

:<math>\nu:x \mapsto (x,x^2,x^3)</math>

That is, it is the closure by a single [[point at infinity]] of the [[affine curve]] <math>(x,x^2,x^3)</math>.

The twisted cubic is a [[projective variety]], defined as the intersection of three [[quadric (algebraic geometry)|quadric]]s. In homogeneous coordinates <math>[X:Y:Z:W]</math> on '''P'''<sup>3</sup>, the twisted cubic is the closed [[scheme (mathematics)|subscheme]] defined by the vanishing of the three [[homogeneous polynomial]]s
:<math>F_0 = XZ - Y^2</math>
:<math>F_1 = YW - Z^2</math>
:<math>F_2 = XW - YZ.</math>
It may be checked that these three [[quadratic form]]s vanish identically when using the explicit parameterization above; that is, substitute ''x''<sup>3</sup> for ''X'', and so on.

More strongly, the [[homogeneous ideal]] of the twisted cubic ''C'' is generated by these three homogeneous polynomials of degree 2.

==Properties==
The twisted cubic has the following properties:
* It is the set-theoretic complete intersection of <math>XZ - Y^2</math> and <math>Z(YW-Z^2)-W(XW-YZ)</math>, but not a scheme-theoretic or ideal-theoretic complete intersection; meaning to say that the ideal of the variety cannot be generated by only 2 polynomials; a minimum of 3 are needed. (An attempt to use only two polynomials make the resulting ideal not [[Radical ideal|radical]], since  <math>(YW-Z^2)^2</math> is in it, but <math>YW-Z^2</math> is not).
* Any four points on ''C'' span '''P'''<sup>3</sup>.
* Given six points in '''P'''<sup>3</sup> with no four coplanar, there is a unique twisted cubic passing through them.
* The [[Union (set theory)|union]] of the [[tangent]] and [[secant line]]s (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 lines are pairwise disjoint, except at points of the variety itself.
* The projection of ''C'' onto a plane from a point on a tangent line of ''C'' yields a [[cuspidal cubic]].
* The projection from a point on a secant line of ''C'' yields a [[Nodal surface|nodal]] cubic.
* The projection from a point on ''C'' yields a [[conic section]].

==References==
*{{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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