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Ideals and Varieties, Rational Normal Cone


May 20, 2012
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 [tex] \phi: \mathbb{C}^2 \rightarrow \mathbb{C}^4 [/tex] defined by [tex] \phi(s, t) = (s^3, s^2t, st^2, t^3) = (x_0, x_1, x_2, x_3) [/tex].

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

I can show that [tex] \phi(\mathbb{C}^2) \subseteq V(I) [/tex] 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?