# Ideals and Varieties, Rational Normal Cone

#### oblixps

##### Member
I have the ideal I = <f1, f2, f3>, where f1 = x0x2-x12, f2 = x0x3 - x2x1, f3 = x1x3 - x22.
I also have the parametrization of some surface given by $$\phi: \mathbb{C}^2 \rightarrow \mathbb{C}^4$$ defined by $$\phi(s, t) = (s^3, s^2t, st^2, t^3) = (x_0, x_1, x_2, x_3)$$.

I want to show that $$V(I) = \phi(\mathbb{C}^2)$$ and I(V) = I = <f1, f2, f3>.

I can show that $$\phi(\mathbb{C}^2) \subseteq V(I)$$ but I can't figure out a way to show the reverse inclusion.

Similarly, I see that <f1, f2, f3> is contained in I(V), but I can't figure out how to show the reverse inclusion either.

Can someone give me some hints on how to proceed in these directions?