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). 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 of a Veronese map of degree three on the projective line.

Definition

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

ν : 𝐏^{1} \to 𝐏^{3}

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

ν : [S : T] \mapsto [S^{3} : S^{2} T : S T^{2} : T^{3}] .

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

ν : x \mapsto (x, x^{2}, x^{3})

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

The twisted cubic is a projective variety, defined as the intersection of three quadrics. In homogeneous coordinates $[X : Y : Z : W]$ on P³, the twisted cubic is the closed subscheme defined by the vanishing of the three homogeneous polynomials

F_{0} = X Z - Y^{2}

F_{1} = Y W - Z^{2}

F_{2} = X W - Y Z .

It may be checked that these three quadratic forms vanish identically when using the explicit parameterization above; that is, substitute x³ 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 $X Z - Y^{2}$ and $Z (Y W - Z^{2}) - W (X W - Y Z)$ , 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, since $(Y W - Z^{2})^{2}$ is in it, but $Y W - Z^{2}$ 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.