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Edit: It seems rude to delete the question, but I have my answer now thanks to rghthndsd.

I'm a bit unsure about the terminology in a homework question I'm doing, and I can't find any clear answers anywhere. So I'm really just here to find out if I've interpreted the question correctly. I don't think I'll have any problems doing the question once I have this settled.

I'm asked to consider $\varphi$, a projection from the point $P=(1:0:0:0)\in\mathbb P^3$ to the hyperplane $\mathscr Z(w)=\{Q\in\mathbb P^3|Q_1=0\}\subset\mathbb P^3$.

I assumed when I saw this that this meant $\varphi$ was a function on $\mathscr Z(w)$, such that $\varphi(Q)$ is the line connecting $P$ and $Q$.

But then the question asks me to consider the Zariski closure of the image of $X\setminus P$ under $\varphi$, where $X=\mathscr Z(x^2-xz-yw,yz-xw-zw)$.

Am I right in thinking that the image in question (prior to finding the Zariski closure) is the collection of lines connecting $P$ to $X\cap\mathscr Z(w)$?

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The projection is a function on $\mathbb{P}^3 \setminus P$. It sends a point $Q \neq P$ to where the (unique) line containing $P$ and $Q$ intersects the hyperplane. Note that the intersections must consist of precisely one point. –  RghtHndSd Mar 29 at 17:04
    
You are a life saver. Thank you. –  user99412 Mar 29 at 17:05
    
I don't understand your last question (in particular, what the "image in question" is), but it sounds like this is a full answer to you. If so, I will make my comment into an answer. –  RghtHndSd Mar 29 at 17:07

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