Twisted cubic

<h2 id="definition">Definition</h2>

The twisted cubic is most easily given <a href="/facts/Parametrically/zFvVFMv9">parametrically</a> as the image of the map

ν
        :
        
          
            P
          
          
            1
          
        
        →
        
          
            P
          
          
            3
          
        
      
    
    {\displaystyle \nu :\mathbf {P} ^{1}\to \mathbf {P} ^{3}}

which assigns to the <a href="/facts/Homogeneous_coordinate/TG9ebKN4">homogeneous coordinate</a> 
 
 
 
 [
 S
 :
 T
 ]
 
 
 {\displaystyle [S:T]}
 
 the value

ν
        :
        [
        S
        :
        T
        ]
        ↦
        [
        
          S
          
            3
          
        
        :
        
          S
          
            2
          
        
        T
        :
        S
        
          T
          
            2
          
        
        :
        
          T
          
            3
          
        
        ]
        .
      
    
    {\displaystyle \nu :[S:T]\mapsto [S^{3}:S^{2}T:ST^{2}:T^{3}].}

In one <a href="/facts/Coordinate_patch/pjoya29t">coordinate patch</a> of projective space, the map is simply the <a href="/facts/Moment_curve/6meK4TPq">moment curve</a>

ν
        :
        x
        ↦
        (
        x
        ,
        
          x
          
            2
          
        
        ,
        
          x
          
            3
          
        
        )
      
    
    {\displaystyle \nu :x\mapsto (x,x^{2},x^{3})}

That is, it is the closure by a single <a href="/facts/Point_at_infinity/GAuS0wHQ">point at infinity</a> of the <a href="/facts/Affine_curve/Vk7f97vn">affine curve</a> 
 
 
 
 (
 x
 ,
 
 x
 
 2
 
 
 ,
 
 x
 
 3
 
 
 )
 
 
 {\displaystyle (x,x^{2},x^{3})}
 
.
The twisted cubic is a <a href="/facts/Projective_variety/aB2XdB7B">projective variety</a>, defined as the intersection of three <a href="/facts/Quadric_(algebraic_geometry)/eCG83ks3">quadrics</a>. In homogeneous coordinates 
 
 
 
 [
 X
 :
 Y
 :
 Z
 :
 W
 ]
 
 
 {\displaystyle [X:Y:Z:W]}
 
 on P3, the twisted cubic is the closed <a href="/facts/Scheme_(mathematics)/c7uSqGqL">subscheme</a> defined by the vanishing of the three <a href="/facts/Homogeneous_polynomial/WUHL7Kct">homogeneous polynomials</a>

F
          
            0
          
        
        =
        X
        Z
        −
        
          Y
          
            2
          
        
      
    
    {\displaystyle F_{0}=XZ-Y^{2}}

F
          
            1
          
        
        =
        Y
        W
        −
        
          Z
          
            2
          
        
      
    
    {\displaystyle F_{1}=YW-Z^{2}}

F
          
            2
          
        
        =
        X
        W
        −
        Y
        Z
        .
      
    
    {\displaystyle F_{2}=XW-YZ.}

It may be checked that these three <a href="/facts/Quadratic_form/l2n1jXPe">quadratic forms</a> vanish identically when using the explicit parameterization above; that is, substitute x3 for X, and so on.
More strongly, the <a href="/facts/Homogeneous_ideal/AIUjkSaP">homogeneous ideal</a> of the twisted cubic C is generated by these three homogeneous polynomials of degree 2.

<h2 id="properties">Properties</h2>
The twisted cubic has the following properties:

<ul><li>It is the set-theoretic complete intersection of 
 
 
 
 X
 Z
 −
 
 Y
 
 2
 
 
 
 
 {\displaystyle XZ-Y^{2}}
 
 and 
 
 
 
 Z
 (
 Y
 W
 −
 
 Z
 
 2
 
 
 )
 −
 W
 (
 X
 W
 −
 Y
 Z
 )
 
 
 {\displaystyle Z(YW-Z^{2})-W(XW-YZ)}
 
, 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 <a href="/facts/Radical_ideal/GYm3Pm9C">radical</a>, since 
 
 
 
 (
 Y
 W
 −
 
 Z
 
 2
 
 
 
 )
 
 2
 
 
 
 
 {\displaystyle (YW-Z^{2})^{2}}
 
 is in it, but 
 
 
 
 Y
 W
 −
 
 Z
 
 2
 
 
 
 
 {\displaystyle YW-Z^{2}}
 
 is not).</li>
<li>Any four points on C span P3.</li>
<li>Given six points in P3 with no four coplanar, there is a unique twisted cubic passing through them.</li>
<li>The <a href="/facts/Union_(set_theory)/KVVXKfWl">union</a> of the <a href="/facts/Tangent/sF0jH3EI">tangent</a> and <a href="/facts/Secant_line/gJnaF5g3">secant lines</a> (the <a href="/facts/Secant_variety/JRC6dQuE">secant variety</a>) of a twisted cubic C fill up P3 and the lines are pairwise disjoint, except at points of the curve itself. In fact, the union of the <a href="/facts/Tangent/sF0jH3EI">tangent</a> and <a href="/facts/Secant_line/gJnaF5g3">secant</a> lines of any non-planar smooth <a href="/facts/Algebraic_curve/LnWsPmDP">algebraic curve</a> is three-dimensional. Further, any smooth <a href="/facts/Algebraic_variety/Vk7f97vn">algebraic variety</a> with the property that every length four subscheme spans P3 has the property that the tangent and secant lines are pairwise disjoint, except at points of the variety itself.</li>
<li>The projection of C onto a plane from a point on a tangent line of C yields a <a href="/facts/Cuspidal_cubic/sR3ITc2B">cuspidal cubic</a>.</li>
<li>The projection from a point on a secant line of C yields a <a href="/facts/Nodal_surface/3SXH9Etl">nodal</a> cubic.</li>
<li>The projection from a point on C yields a <a href="/facts/Conic_section/teqgLptX">conic section</a>.</li></ul>

<ul><li>Harris, Joe (1992), Algebraic Geometry, A First Course, New York: Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-97716-3.</li></ul>

Twisted cubic open-in-new

Twisted cubic