Let $X$ be an $A$-scheme ($A$ a ring) and $\mathscr{L}$ be a line bundle on $X$. Define the ring of sections of $\mathscr{L}$ to be $R(X,\mathscr{L})=\bigoplus _{n\geq 0} \Gamma(X,\mathscr{L^{\otimes n}})$, and let $Y=Proj(R(X,\mathscr{L}))$.
Apparently if the ring is finitely generated and $\mathscr{L}$ corresponds to an effective divisor then there is a rational map $X \dashrightarrow Y$, how is this map defined? What do we use the finite generation of the ring of the effectiveness for?
And if $\mathscr{L}$ is ample then $Y$ is the image of the map defined by $\mathscr{L}$, $X\rightarrow \mathbb{P}_A^n$, why is this?
Update: As one comment mentioned this is related to Hartshorne II.7.1, I am aware of this correspondence between n+1 generating global sections of a line bundle and maps to $\mathbb{P}^n_A$, which is what I referred to when I said "the map defined by $\mathscr{L}$, $X\rightarrow \mathbb{P}_A^n$". I don't know how to use the construction one usually does to get map to projective space using sections to show that the image of the map given by $\mathscr{L}$ lands in $Y$.