I've been doing some reading on divisors (Chapter 4 of Murty's book on Abelian Varieties) and I have a fairly elementary question I've spent far too much time trying to figure out. Many authors treat the equivalence between (certain) linear systems and projective embeddings. Going from a linear system, one chooses a basis of $\mathcal{L}(D)$ and defines a basis.
But it is unclear to me how to go about to get the divisor of projective embedding. Let $\varphi = [\varphi_0:\varphi_1:\cdots : \varphi_m]$. Let's write the (Weil) divisor of a meromorphic function as $(f) = (f)_0 -(f)_\infty$. Now, the book writes $$ 0 \leq \left(\frac{\varphi_i}{\varphi_0} \right)_0 = \left(\frac{\varphi_i}{\varphi_0} \right)+\left(\frac{\varphi_i}{\varphi_0} \right)_\infty $$ If we set $D_i = \left(\frac{\varphi_i}{\varphi_0} \right)_\infty$, then the claim is that $D_1,\dots, D_m$ are all linearly equivalent. But I don't see that. Further, is there a canoncial choice of effective divisor given $\varphi$.