Twisted cubic

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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.

[edit] Definition

It is most easily given parametrically as the image of the map

\nu:\mathbb{P}^1\to\mathbb{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

\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,x2,x3).

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

F0 = XZ − Y2
F1 = YW − Z2
F2 = XW − YZ

It may be checked that these three quadratic forms vanish identically when using the explicit parameterization above; that is, substituting S3 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 − Y2,YW − Z2,XW − YZ}.

[edit] Properties

The twisted cubic has an assortment of elementary properties:

  • It is the set-theoretic complete intersection of XZ − Y2 and X2Z − XW2 − XY2 + YW2 + YZW − Z3, but not a scheme-theoretic or ideal-theoretic complete intersection.
  • 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.

[edit] References

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