Question about the ideal $I=(xy,yz,zx)$ in the ring $\mathbb C[x,y,z]$.

Question

Given the ideal $I=(xy,yz,zx)$ in the ring $\mathbb C[x,y,z]$. I want to compute $V(I)$, which is the intersection of all ideals containing $I$. And I also want to prove that $I$ can't be generated by two elements over $\mathbb C[x,y,z]$. How can I do this?

Answer 1

Hint for #1: We know, for example, that $xy\in I$. If $P$ is a prime ideal containing $I$, we can deduce something very useful about $P$.

Hint for #2: Notice that the vector space of degree $2$ polynomials in $I$ is $3$-dimensional, so it cannot be generated by $2$ elements. Use this fact to derive a contradiction if $I$ can be generated as an ideal by $2$ elements.