Let $K$ be an algebraically closed field (assume characteristic zero if you like.) Consider the set $S$ in $K^3$ parametrised by $(X,Y,Z)=(t^3,t^4,t^5)$. We see that $S$ is the simultaneous zero set of the ideal $I=(X^3−YZ,Y^2−XZ,Z^2−X^2Y):=(f,g,h).$
How can I show that $I$ cannot be generated by 2 elements? I think it might be useful to note that:
1. $I$ is prime.
2. $f,g,h$ are linearly independent over $K$.
3. $f,g,h$ are all irreducible.
Any ideas?
