Twisted cubic

In mathematics, a twisted cubic is a smooth, rational curve C of degree three in projective 3-space P³. 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 is not 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.

Definition

It is most easily given parametrically as the image of the map

$\nu :\mathbf {P} ^{1}\to \mathbf {P} ^{3}$

which assigns to the homogeneous coordinate $[S:T]$ the value

$\nu :[S:T]\mapsto [S^{3}:S^{2}T:ST^{2}:T^{3}].$

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

$\nu :x\mapsto (x,x^{2},x^{3})$

That is, it is the closure by a single point at infinity of the affine curve $(x,x^{2},x^{3})$.

Equivalently, it is a projective variety, defined as the zero locus of three smooth quadrics. Given the homogeneous coordinates [X:Y:Z:W] on P³, it is the zero locus of the three homogeneous polynomials

$F_{0}=XZ-Y^{2}$

$F_{1}=YW-Z^{2}$

$F_{2}=XW-YZ.$

It may be checked that these three quadratic forms vanish identically when using the explicit parameterization above; that is, substituting x³ for X, and so on.

In fact, the homogeneous ideal of the twisted cubic C is generated by three algebraic forms of degree two on P³. The generators of the ideal are

$\{XZ-Y^{2},YW-Z^{2},XW-YZ\}.$

Properties

The twisted cubic has an assortment of elementary properties:

• It is the set-theoretic complete intersection of XZ-Y² and $Z(YW-Z^{2})-W(XW-YZ)$, but not a scheme-theoretic or ideal-theoretic complete intersection (the resulting ideal is not radical, since $(YW-Z^{2})^{2}$ is in it, but $YW-Z^{2}$ is not).
• Any four points on C span P³.
• Given six points in P³ with no four coplanar, there is a unique twisted cubic passing through them.
• The union of the tangent and secant lines, the secant variety, 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 tangent and 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 P³ 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 cubic.
• The projection from a point on C yields a conic section.

References

• Harris, Joe (1992), Algebraic Geometry, A First Course, New York: Springer-Verlag, ISBN 0-387-97716-3.