I consider a Hirzebruch surface $\mathbb{F}_2$, with a unique $(-2)$-curve denoted by $Q$ and I denoted the ruling by $P$ which is $0$-curve. Now I consider divisor of the form $D=Q+P$ and consider the line bundle $L=O(Q+P)$. Then I consider the following exact sequence: $$ 0\rightarrow O(-Q-P)\xrightarrow{j} O^{\oplus 2}\rightarrow coker(j)\rightarrow 0 $$
I would like to know what does this $coker(j)$ look like. I think it should be a sheaf with a torsion subsheaf supported on $Q$ and the corresponding quotient sheaf should be a locally free sheaf, in particular, it should be a line bundle, but I dont know what does it look like
It might be very trivial Thanks
The map $j$ factors through $O(-Q-P)\to O(-P)\to O^2$. The cokernel of $O(-P)\to O^2$ is just $O(P)$. The cokernel of $O(-Q-P)\to O(-P)$ is $O_Q(-1)$. So, you have an exact sequence, $0\to O_Q(-1)\to C\to O(P)\to 0$, where $C$ is the cokernel of $j$.