Let $X$ be an elliptic curve with a fixed point $P_0$. I read two different definitions about group structure on it. One is to identify $X$ with its Jacobian (i.e. $Pic^0(X)$). we can consider the map $P\mapsto \mathcal L(P-P_0)$ which is obviously injective and also surjective by Riemann-Roch. There is another geometric definition, i.e. $P+Q+R=0$ if and only if they are collinear. I want to know how to see these two definitions coincide with each other?
In fact I have found an explanation on Hartshorne (page 321). But I don't understand this line:
If we embed $X$ in $\mathbb P^2$ by the linear system $|3P_0|$, then three points $P,Q,R$ of the image are collinear if and only if $P+Q+R\sim3P_0.$
Thanks in advance.
Let $D$ be a very ample effective divisor. The map $\phi_D : X \to \Bbb P(H^0(X,D))$ is constructed from a basis $f_0, \dots, f_m$ of $H^0(X,D)$ and we set $\phi_D(x) = [f_0: \dots : f_m]$. By construction there is a $1-1$ correspondence between effective divisors $E \sim D$ and hyperplane sections of $\phi_D(X)$.
Indeed if $E \sim D$ is effective, we have $E - D = \text{div}(g)$ for some $g \in K(X)$. By definition $g \in H^0(X,D)$ since $\text{div}(g) + D = E \geq 0$, so $g = a_1f_1 + \dots + a_mf_m$ corresponds to the hyperplane section $H_E : a_0f_0 + \dots + a_mf_m = 0$.
Conversely, any such hyperplane section give a section $g \in H^0(X,D)$ and an effective divisor $E_H = \text{div}(g) + D$.
In particular, we have $E := P+Q+R \sim 3P_0$ if and only $E$ is an hyperplane section of $X$, that is $P,Q,R$ are collinear.