Singular Book 1.8.6 -- Zariski closure of the image

We compute an implicit equation for the surface defined parametrically by the map f : A² → A³, (u,v) → (uv,uv²,u²).

i1 : loadPackage "Elimination";

i2 : A = QQ[u,v,x,y,z];

i3 : I = ideal "x-uv,y-uv2,z-u2"

                            2         2
o3 = ideal (- u*v + x, - u*v  + y, - u  + z)

o3 : Ideal of A

i4 : eliminate(I,{u,v})

            4    2
o4 = ideal(x  - y z)

o4 : Ideal of A

This ideal defines the closure of the map f, the Whitney umbrella.

Alternatively, we could take the coimage of the ring homomorphism g corresponding to f.

i5 : g = map(QQ[u,v],QQ[x,y,z],{x => u*v, y => u*v^2, z => u^2})

                                       2   2
o5 = map(QQ[u, v],QQ[x, y, z],{u*v, u*v , u })

o5 : RingMap QQ[u, v] <--- QQ[x, y, z]

i6 : coimage g

     QQ[x, y, z]
o6 = -----------
        4    2
       x  - y z

o6 : QuotientRing