I'm studying Fulton's algebraic curves book and some of its notation look like confusing.
I need help to understand this equivalence in the page 35.
The preceding paragraph he mentioned I put below:
If $G$ is a form in $k[X,Y,Z]$ of degree $d$, then $G_*=\frac{G}{L^d}$ is an element of $k(\mathbb P^2)$, because with the natural identification we have $\Gamma (\mathbb P^2)=\Gamma(\mathbb A^2)=k[X,Y,Z]/I(\mathbb A^2)=k[X,Y,Z]/(0)=k[X,Y,Z]$. Then $k(\mathbb P^2)=\{\frac{f_1}{f_2}|f_1,f_2\in k[X,Y,Z]\}$.
Thus, $\overline G_*=\frac{\overline G}{\overline {L^d}}, \text{where}\ \overline G, \overline {L^d}\in \Gamma (F) \ \text{and}\ \overline {L^d}(P)\neq 0$.
If I'm right so far, why $\text{ord}_P^F\bigg(\frac{\overline G}{\overline {L^d}}\bigg)$ and $\text{ord}_P^F\bigg(\frac{G}{H}\bigg)$ (defined in the highlighted part) are equivalent?
I need help
Thanks in advance
EDIT
It seems the author want to turn this form $G$ into a rational function, why? a form in particular is a rational function itself.
I think the point is that if $G$ is a form of degree $n$, then you can turn it into a rational function of degree $0$ by dividing by another form of degree $n$. On course if you now want the order at a point $P$, you should divide by a form that doesnt pass through $P$. Fulton suggests $L^n$ but his remark is that any such form say $H$ would do as well. ord is additive so
$$ord (G/L^n)=ord([G/H][ H/L^n])=ord(G/H)+ord(H/L^n)=ord(G/H)$$ becsuse $H/L^n$ has no zero or pole at $P$.