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I need to write the ideal $I= (xz-y^2, yt- z^2) \subset R = \mathbb{K}[x,y,z,t]$ as intersection of prime ideals. Any idea?

For the moment, I've noticed that $I$ is radical, then it suffices to find the minimal ideals associated to $I$. In addition, I have \begin{equation} x(yt -z^2) + z(xz -y^2) = xyt -zy^2 = y(xt -yz) \in I \end{equation} If $\mathfrak p \in \operatorname{Min}(I)$, then $y \in \mathfrak p$ or $xt-yz \in \mathfrak p$.

If $y \in \mathfrak p$, then $z^2 \in \mathfrak p$ and then $z \in \mathfrak p$. It follows $\mathfrak p=(y,z)$.

If $xt−yz \in \mathfrak p$, I consider $J=(I,xt−yz)$ and I want to prove that $J$ is prime. A strategy could be demonstrating that $J$ is $\ker \psi$, where $ψ:\mathbb{K}[x,y,z,t]\to\mathbb{K}[u]$. Now, I am not able to follow.

user26857
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Ella Smith
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Actually $J$ is the kernel of the map $K[x,y,z,t]\to K[u,v]$ given by $x\mapsto u^3$, $y\mapsto u^2v$, $z\mapsto uv^2$, $t\to v^3$. (See also here.)

Now you find that $I=(y,z)\cap J$, an intersection of two prime ideals.

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