Intersection of complex sphere and cone

[huyichen]

Show the intersection of complex sphere (|z1|^2+|z2|^2+|z3|^2=1) in C^3 and the complex cone (z1^2+z2^2+z3^3=1) in C^3 is a smooth submanifold of C^3.

I am trying to do it using regular level set, but I am not sure which one of (1,0) or (1,1,0) should be set to be the regular value?

 

[zhentil]

You have a map from C^3 to R x C. The intersection is the preimage of (1,1,0).