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

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

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

which assigns to the homogeneous coordinate ${\displaystyle [S:T]}$ the value

${\displaystyle \nu$ :[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

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

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

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

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

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

${\displaystyle F_{2}=XW-YZ.}$

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

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

Properties

The twisted cubic has an assortment of elementary properties:

• It is the set-theoretic complete intersection of ${\displaystyle XZ-Y^{2}}$ and ${\displaystyle Z(YW-Z^{2})-W(XW-YZ)}$, but not a scheme-theoretic or ideal-theoretic complete intersection (the resulting ideal is not radical, since ${\displaystyle (YW-Z^{2})^{2}}$ is in it, but ${\displaystyle YW-Z^{2}}$ 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.