# How to find the equation of an image under a central projection

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!

• First, $(s,t)\in \mathbb{P}^1$ not 3 as you write. $\pi(C)$ is a curve in $\mathbb{P}^2$, so you need to find an irreducible homogeneous polynomial $f(x_1,x_2,x_3)$ such that $f(s^3,st^2,t^3)=0$ for all $(s:t)$. – Mohan Jan 13 at 19:07
• so $f=x_1x_3^2-x_2^3$ would do the job right? Is that really the whole exercise, it seems too simple – get rekt m8 Jan 13 at 19:17
• @Mohan could you tell me what I would have to do if $P$ was a different point? I was told I need a change of coordinates, but i am not sure how that works – get rekt m8 Jan 13 at 19:57
• What does $V(x_2)$ mean? – amd Jan 13 at 20:32
• @amd the subset of $\mathbb{P}^3$ where the second coordinate is $0$. So $z \in V(x_2)$ has the form $(a:0:c:d)$ where not every coordinate is zero. – get rekt m8 Jan 13 at 20:50