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 which assigns to the homogeneous coordinate [S:T] the value In one coordinate patch of projective space, the map is simply the moment curve 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 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: