Suppose we have a map $f:\mathbb{P}^n\rightarrow \mathbb{P}^m$ which is algebraic. What are the techniques to find the equations defining the image of f as a subvariety of $\mathbb{P}^m$?
For example in the case $f:\mathbb{P}^1\rightarrow \mathbb{P}^2$ , $[x,y]\mapsto [x^3,x^2y,y^3]$. I immediately see the equation $z_0^2z_2-z_1^3=0$ is satisfied on the image, and I suspect strongly that this is the only equation needed as in Euclidean coordinates we get a curve (cusp).