Equations of a projective variety from parametric ones

Question

How does one find equations of a variety given parametric equations (i.e. a regular map) in projective space? For example, I got stuck in finding the equations of the curve in $\Bbb{P}^2$ described by $$\begin{align}z_0 & =u^3-v^3 \\ z_1 & =u^2v \\ z_2 &= uv^2 \end{align}$$ where $[u,v]\in\Bbb{P}^1$. I have just been playing around with it without getting anywhere. I am wondering if there are any general methods for this kind of problem.

Answer 1

I don't know of a general method, but this case seems fairly nice: We have $v = z_1/u^2$ so $u^3 = z_1^2/z_2$. Using both of these in the first equation gives the equation for the variety in $\mathbb{P}^2$ as $$ z_0z_1z_2 = z_1^3 - z_2^3.$$

Answer 2

The dimension of the space of homogeneous polynomials of degree $d$ in the $z_i$ is $(d+1)(d+2)/2$, while the dimension of the space of homogeenous polynomials of degree $3d$ in $u,v$ is $3d+1$

If you pick $d=4$ the first space is $15$-dimensional and the second is $13$-dimensional, thus there must be a nontrivial polynomial of degree $4$ in the $z_i$ that vanishes after doing the substitution.

(though actually as Ahsan shows there is one of degree $3$)