I'd like to understand the following claim.
Claim:
Consider a projective manifold $X=V(F_1,F_2,\cdots,F_l)$ in complex projective space $P_n(\mathbb{C})$.
Let $P=(a_0,a_1,\cdots,a_n)$ be a fixed point outside $X$, and let $H:\sum_{i=0}^n{\alpha_ix_i}=0$ be a fixed hyperplane that does not pass through $P$.
We choose $H$ to be as general as possible.
Given a point $Q=(b_0,b_1,\cdots,b_n)$ of $X$, we consider the line $\overline{PQ}$ joining $P$ and $Q$. Using homogeneous parameters $s:t$, we can parameterize $\overline{PQ}$ as
$(a_0s+b_0t:a_1s+b_1t:\cdots:a_ns+b_nt),(s:t)\in P^1(\mathbb{C})$.
The line $\overline{PQ}$ intersects $H$ at a unique point. This is clear since the equation $\sum_{i=0}^n{\alpha_i(a_is+b_it)}=0$ has a unique solution $(s_0:t_0)$. We denote this intersection point by $R(Q)$. Thus, we define a map $\phi_P:X\to H$ by $Q\mapsto R(Q)$. Since $H$ can be identified with complex projective space $P^{n-1}(\mathbb{C})$, we can regard $\phi_P(X)$ as a subset of $P^{n-1}(\mathbb{C})$. Actually, $\phi_P(X)$ is again a projective manifold. We call $\phi_P$ the projective transformation centered at $P$. If $\phi_P(X)\neq H$, we can repeat the same process for $X_1=\phi_P(X)$, centering at a point $P_1$ in $H=P^{n-1}(\mathbb{C})$ to obtain a projection
$\phi_{P_1}:X_1\mapsto P^{n-2}(\mathbb{C})$.
Repeating this process several times, we finally obtain a projection
$\phi_{P_m}:X_m\mapsto P^{n-m-1}(C)$ which is surjective. Moreover, the dimension of $X$ is $n-m-1$
This is the claim. What I am looking for is:
1.A textbook where the proof of this proposition is written.
2.Whether this proposition can be interpreted in terms of commutative algebra, for example, in terms of projective/injective resolutions, Koszul complexes,hilbert polynomial,etc.
If there are any other interesting facts, please let me know.
What’s going on here in commutative algebra is basically (graded) Noether normalisation. Each of your maps is is finite onto its image (all the fibres are finite sets, and can be proven algebraically), and this process terminates to yield a finite surjective map onto a projective space. The dimension claim then follows since finite maps between irreducible things preserve dimension (a fact that can be seen in a few different ways, depending on your definition of dimension).
This process is also a good way to think about (and prove) Noether normalisation in the affine case, taking an arbitrary projective closure and picking your projection points carefully yields the desired finite map to $\mathbb{A}^k$/system of parameters for the algebra.
I’m not sure of a source that proves Noether normalisation in this manner, but once you’re comfortable with finite maps, this is a fantastic exercise to do yourself.