Monomial curve with exponents three, four and five ($k[t^3,t^4,t^5]$)

The kernel of $x\mapsto t^3$, $y\mapsto t^4$, $z\mapsto t^5$ is generated by $y^2-xz$, $yz-x^3$, $z^2-x^{2}y$. Reducing these three relations leaves a unique normal form $A_0(x)+A_1(x)y+A_2(x)z$: its images have distinct exponents modulo three. The curve ring is therefore finite free of rank three over $k[x]$ and has dimension one. In a three-variable polynomial ring its ideal has height two, but its three independent quadratic initial forms force at least three generators. This is a concrete failure of generation by codimension many equations.