Exact Sequence of Line Bundles on $\mathbb{P}^{2}$ I'm considering an example in the great book "Mirror Symmetry" where they consider the exact sequence of line bundles 
$\mathcal{O}(-2) \to \mathcal{O}(-1) \oplus \mathcal{O}(-1) \to \mathcal{O}$, 
such that the first map is given by $s \mapsto (gs,fs)$ ($f,g$ are fixed linear functions on $\mathbb{P}^{2}$), and the second map is given by $(s_{1},s_{2}) \mapsto fs_{1}-gs_{2}$.  Of course, these $s$ are sections of the bundle.  
My confusion is how these sections even produce a well-defined sequence.  On $\mathbb{P}^{2}$ I think neither $\mathcal{O}(-2)$ nor $\mathcal{O}(-1)$ admit global sections.  So the sections above are meant to be interpreted (I think) as simply holomorphic functions on some open set.  But the above map seems to be saying that if $s$ is a section of $\mathcal{O}(-2)$, then $gs$ will be a section of $\mathcal{O}(-1)$.  How does this work?  I know the transition functions of $\mathcal{O}(-2)$ must be the square of those for $\mathcal{O}(-1)$, but I'm incredibly confused as to how that's consistent with the definition of these maps.  Thanks for any tips here!
 A: This is not an answer, but too long for a comment.
One way to think about this, is to think of $\mathscr O_{\mathbb P^2}(d)$ as the $d$th graded part of $k[x_0,x_1,x_2]$ (and if $d < 0$, think of it as $Hom(k[x_0,x_1,x_2]_d,k[x_0,x_1,x_2]_0)=Hom(k[x_0,x_1,x_2]_d,k)$). Then the maps in the sequence
$$
\mathscr O(-2) \to \mathscr O(-1)^2 \to \mathscr O \to 0
$$
can be identified with the maps
$$
hom(k[x_0,x_1,x_2],k) \xrightarrow{\varphi} hom(k[x_0,x_1,x_2],k)^2 \xrightarrow{\psi} k[x_0,x_2,x_2],
$$
defined as follows. The first map sends a linear functional $\lambda$ to linear functional defined as $\varphi(\lambda)(h,h')=(fh,gh')$, where $h,h'$ are polynomials in $k[x_0,x_1,x_2]$. The map is of degree $1$, since it increases degrees. 
Similarly, $\psi$ is given by sending a pair of funtionals $(\lambda,\lambda')$ to the functional given by $f \mapsto \lambda(f)-\lambda'(g)$.
(my point here is really to think of the structure sheaf of $\mathbb P^2$ as a graded module, which makes it easier to think of sections). 
Also note that a map of sheaves does not need to be induced by a map of global sections (can only be true if the sheaves are globally generated).
(unrelated: which book on Mirror Symmetry?)
