Curves of degree three ("the twisted cubic").

Every irreducible, non degenerated (i.e. not contained in a plane) curve, C, of degree 3 in P³ is smooth (otherwise a plane through a singular point and two other points on the curve will intersect the curve in > 3 points); such a curve is also rational: let R be a bisecant to C, the pencil a planes through R gives a parametrization of C. So our curve is isomorphic to P¹. Since there is only one linear system of degree 3 and (projective) dimension 3 on P¹, it turns out that any two smooth space curves of degree 3 are projectively equivalent. For this reason one usually speaks of THE twisted cubic.

The twisted cubic is projectively normal. This can be seen as follows: consider the exact sequence of restriction to a plane H:

0 -> I_C(k) -> I_C(k+1) -> I_CH(k+1) -> 0

Since h¹(I_C(1))=0, it is enough to show that h¹(I_CH(k+1))=0 for k > 1, and this follows easily since CH is a set of 3 points in general position in the plane H.

In particular I_C(2) is generated by global sections and C lies on a smooth quadric surface (this can be proved directly observing that the quadric cones containing C are precisely the cones of base C with vertices a point on C; so there are 00¹ singular quadrics containing C. Since there are 00² quadrics containing C, the general one is smooth. See also Castelnuovo-Mumford's lemma.). So C is a divisor of type (1,2) on a smooth quadric surface. The minimal free resolution of the graded ideal is given by the minimal free resolution of a curve on a quadric or by the minimal free resolution of 3 points in P² in general position (since the curve is a projectively normal curve, its m.f.r. is the same, numerically, as the m.f.r. of its general (any) plane section).