Coordinate projection in projective space

Question

I was thinking about this:
Consider the affine variety on an algebrically closed field $K$ given by $A:=\frac{K[t,y]}{(xt-1)}$ and the projection on the $t$ coordinate. This is a morphism of affine scheme $f:Spec(A)\longrightarrow Spec(K[t])$ and if $\phi:K[t]\longrightarrow A$ is the ring map which maps $t$ to its class we have $Spec(\phi)=f$. Now it is possible to get the fiber from the morphism $f$.
What happen if i take the projective closure $\tilde A$ given by $Y:=Proj(\frac{K[x,y,t]}{(xt-y^2)})$ how can i construct the map on $\mathbb{P}^1$?
Is there any corrispondence between the graded rings using $Proj$ functor?
My idea is to consider a line in $\mathbb{P}^2$ and intersect it with the variety $Y$ but i am not able to write down the details, any suggestion or reference please?

Answer 1

This is called "projection from a point on to a hyperplane", and doing this in $\Bbb P^n$ is a common technique in algebraic geometry. Given a point $P$ and a hyperplane $H$ not containing $P$, both in $\Bbb P^n$, given any point $Q\neq P$ the line $\overline{PQ}$ intersects $H$ in exactly one point $P'$. We can then define a map $\Bbb P^n\setminus\{P\} \to H$ given by sending $Q\mapsto P'$. The reason this is called projection is because if we choose coordinates so that $P=[0:\cdots:0:1]$ and $H=V(x_n)$, then the morphism sends $[a_0:\cdots:a_{n-1}:a_n]\mapsto [a_0:\cdots:a_{n-1}:0]$.

If you're interested in more material, I don't have a specific book/text/notes recommendation, but searching for "projection from a point" or "projection away from a point" will likely get you a fair amount of material. (Cox, Little, and O'Shea's Ideals, Varieties, and Algorithms does contain a fair amount of computational tools for computing projections via elimination theory and resultants, but I've never tried learning about or teaching projection through this book, so I'm not sure how suitable it would be for an introduction.)

Your question about a relationship between the coordinate algebras because of the "functor" proj is a little misleading. Proj isn't a functor, and the projection map isn't defined on all of $\Bbb P^n$ anyways. You can always look locally on any affine open $U$ not containing $P$ to get a projection map $U\to U\cap H$ as in the affine case, and this is good enough to do a lot of work with.