Let $Y$ be the curve given parametrically by $x=t^3, y = t^4, z = t^5$. Show that $I(Y)$ is a prime ideal of height $2$ in $k[x,y,z]$ which cannot be generated by two elements.
Obviously $(x^4-y^3,x^5-z^3) \subseteq I(Y)$, but these ideals can't be equal because the one on the right is generated by two elements. So, what is the prime $I(Y)$? And how can i show its height is precisely 2?
Well, implicitization is given by taking the ideal $I=\langle x-t^3,y-t^4,z-t^5\rangle$ w.t.r. the lex ordering $t>x>y>z$ and eliminating $t$ by considering the first elimination ideal $I_1 = I\cap k[x,y,z]$, which is given by the elimination theorem via its Gröbner basis $G_1 = G\cap k[x,y,z]$ where $G$ is a Gröbner basis of $I$. I've performed the calculation in Singular giving $G$ as follows:
$y^5-z^4,xz-y^2,xy^3-z^3, x^2y-z^2, x^3-yz, tz-x^2,ty-z, tx-y, t^3-x$.