Let $\pi:\mathbb{P}^3 \to V(x_2) \cong \mathbb{P}^2$ the linear projection with center $P =(0:1:0:0)$. Find the equation for the image of $C=\{(s^3:s^2t:st^2:t^3)|~(s:t) \in \mathbb{P}^1 \}$ under $\pi$.
I am currently trying to solve this problem.
What I know is that $\pi(C)=\{(s^3:0:st^2:t^3)|~(s:t) \in \mathbb{P}^3 \}$. However I am uncertain of what is meant by 'Find the equation for the image of $C$'. My guess is that I need to find a polynomial $f \in k[x_1,x_2,x_3,x_4]$ such that $a\in C \Leftrightarrow f(a)=0$. Is that correct? And how do I find that?
Any help would be appreciated!