Twisted Cubic - Properties

Properties

The twisted cubic has an assortment of elementary properties:

• It is the set-theoretic complete intersection of XZ-Y2 and, but not a scheme-theoretic or ideal-theoretic complete intersection (the resulting ideal is not radical, since is in it, but 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.