I have been reading a proof of the following proposition from a set of lecture notes, the statement is as follows.
Let $I \subset S = \mathbb{C}[x_0,...,x_n]$ be a homogeneous ideal, where $H_{S/I}(r) = d$ for all $r$ sufficiently large. Then $V(I)\subset \mathbb{P}^n$ is a finite variety.
Here $H_{S/I}$ denotes the Hilbert function, that is $H_{S/I}(k) = \operatorname{dim}_\mathbb{C}\left(( \mathbb{C}[x_0,...,x_n]/I)_k\right)$. The proof begins as follows:
Assume first that $|V(I)| = \infty$, then we may fix $d+1$ points $p_1,...,p_{d+1}$ such that the first coordinate $p_i^{(0)}$ of each point $p_i$ is non-zero. Furthermore, we may assume without loss of generality that each of the ratios $p_i^{(1)}/p_i^{(0)}$ is distinct.
I can understand the validity of the first of these assumptions as we can use the Vandermonde matrix to force this to be true, if it is not true ahead of time. However, I am not convinced of the second assumption. I am looking for a way to understand why the second condition may be considered to be true without loss of generality.
I have considered the three points $(1:2:2),(1:2:3),(1:3:3) \in \mathbb{P}^2$, we can see that the ratios $p_i^{(1)}/p_i^{(0)}$ are not distinct and neither are $p_i^{(2)}/p_i^{(0)}$, even $p_i^{(2)}/p_i^{(1)}$ are not distinct! So surely I have misunderstood what "without loss of generality" means in this context.
My Question:
What does without loss of generality mean in the context provided here, and why does my example not invalidate this claim? (I should make precise here that I am not claiming I have a counter-example, instead I am wondering why this is not a counter-example)
I have been able to answer to my own question and am posting my answer to help any who may have the same question in the future.
Suppose that we take $d$ points in $V(I)$ all of which have first coordinate scaled to $1$, ie $p_i^{(0)} = 1$ for $i=1,...,d$. Now suppose for the sake of contradiction that for all choices of $p_{d+1} \in V(I)$ we have a repeated ratio $p_i^{(j)}/p_i^{(0)}$ for some $j$, then this means that we have a maximum of $d^n$ points in $V(I)$, contradicting the fact that $|V(I)| = \infty$. This is because we must have a maximum of $d$ choices in each of the positions $p^{(1)},...,p^{(n)}$.
Therefore there must exist some $j$ such that all of the ratios $p_i^{(j)}/p_i^{(0)}$ are distinct, and so we may suppose that $j=1$, as otherwise we can perform a change of coordinates, or some reflection, to interchange each of the $p_i^{(j)}$ and $p_i^{(1)}$.