equations of the twisted cubic

In this example we create the ring homomorphism corresponding to the parametrization of the twisted cubic and compute its kernel, obtaining the equation in P³ of the twisted cubic.

i1 : R = QQ[s,t]

o1 = R

o1 : PolynomialRing

i2 : S = QQ[w,x,y,z]

o2 = S

o2 : PolynomialRing

i3 : f = map(R,S,{s^3,s^2*t,s*t^2,t^3})

               3   2      2   3
o3 = map(R,S,{s , s t, s*t , t })

o3 : RingMap R <--- S

i4 : kernel f

             2                    2
o4 = ideal (y  - x*z, x*y - w*z, x  - w*y)

o4 : Ideal of S