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 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.

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^2T: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 .