Every rational quartic curve is contained in an unique quadric surface. Indeed taking the cohomology in the exact sequence:
0 -> IC(2) -> OP(2) -> OC(2) -> 0
we get h0(IC(2)) > 0. Since the genus of a complete intersection (2,2) is one (in general the genus of a complete intersection (a,b) is 1+ab(a+b-4)/2), the only possibility is h0(IC(2)) = 1.
The quadric surface containing a rational quartic curve is smooth (Ex.1)
In conclusion C is a curve of bidegree (1, 3) on a smooth quadric surface Q.
The curve C has 001 (="one infinity of") trisecants (Q is the surface of trisecants to C; through any point of C passes exactly one trisecant to C).
The general plane section of C consists of 4 points in general position (complete intersection of two conics; notice however that C is not the complete intersection of two quadrics).
If G = C.H is not in general position, then H contains a trisecant to C (H is a tangent plane to Q). In this case the minimal free resolution of the plane section G is:
0 -> O(-4)+O(-3) -> O(-3)+O(-2) -> IG -> 0.
The homogeneous ideal, I(C), is generated in degrees <= 3 (IC is 3-regular by Castelnuovo-Mumford).In particular every rational quartic lies on a smooth cubic surface.
The homogeneous ideal, I(C) is generated by the quadric, Q, and by 3 cubics. The minimal free resolution is given by the minimal free resolution of a curve on a quadric.