Twisted cubic curves

A twisted cubic curve is a rational normal curve in P3, i.e. a rational curve of degree 3 that does not lie in a plane. An example is the image of the map [ x : y ] -> [ x^3 : x^2 y : x y^2 : y^3]. The wikipedia page on twisted cubics has many interesting properties.

Properties

 * Any two twisted cubics are related by a projective change of coordinates.
 * Furthermore, any two parametrized twisted cubics are related by a projective change of coordinates.

Natural bundles and sheaves

 * The normal bundle to a twisted cubic is a rank two vector bundle on P1, hence splits by the Birkhoff-Grothendieck theorem. It is isomorphic to O(5) \oplus O(5).

Moduli

 * There is a 12-dimensional family of twisted cubics. It can be constructed as a quotient of an open subset of P15 (the projective space of 4-tuples of cubic polynomials in x and y) by the action of PGL(2).
 * There are natural compactifications of this space, such as the Hilbert scheme H... or the moduli stack M... of genus-zero unmarked stable maps to P3 of degree 3.

Generalizations

 * Rational normal curves
 * Curves in P3