Ideals and Varieties, Rational Normal Cone

[oblixps]

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?

 

[jvicens]

Sure, I can definitely provide some guidance on how to proceed in these directions.

First, let's recall the definitions of V(I) and I(V):

- V(I) is the set of all points in the complex plane that satisfy all the equations in the ideal I. In this case, since I = <f1, f2, f3>, V(I) is the set of all points (x0, x1, x2, x3) that satisfy the equations f1 = 0, f2 = 0, and f3 = 0.

- I(V) is the ideal of all polynomials that vanish on the set V. In this case, since V = \phi(\mathbb{C}^2), I(V) is the set of all polynomials that vanish on the surface defined by \phi(s, t).

Now, to show that \phi(\mathbb{C}^2) \subseteq V(I), we need to show that for any point (s, t) in \mathbb{C}^2, the corresponding point (s^3, s^2t, st^2, t^3) satisfies the equations f1 = 0, f2 = 0, and f3 = 0.

To do this, we can simply plug in the values of x0, x1, x2, x3 into the equations f1, f2, and f3 and show that they all equal 0. For example, let's look at f1:

f1 = x0x2 - x1^2 = (s^3)(st^2) - (s^2t)^2 = s^5t^2 - s^4t^2 = s^2t^2(s^3 - s^2) = 0

Similarly, you can show that f2 and f3 also equal 0 when we plug in the values of x0, x1, x2, x3. Therefore, we have shown that for any point (s, t) in \mathbb{C}^2, the corresponding point (s^3, s^2t, st^2, t^3) satisfies the equations f1 = 0, f2 = 0, and f3 = 0, which means that \phi(\mathbb{C}^