Twisted cubic

In mathematics, a twisted cubic is a smooth, rational curve $C$ of degree three in projective 3-space $mathbb\{P\}^3$. 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

:$u:mathbb\{P\}^1\; omathbb\{P\}^3$

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

:$u:\; [S:T]\; mapsto\; [S^3:S^2T:ST^2:T^3]\; .$

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

:$u:x\; mapsto\; (x,x^2,x^3)$

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 $mathbb\{P\}^3$, 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 $S^3$ 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 $mathbb\{P\}^3$. 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 $XZ-Y^2$ 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 $mathbb\{P\}^3$.
* Given six points in $mathbb\{P\}^3$ 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 $mathbb\{P\}^3$ 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 $mathbb\{P\}^3$ 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

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