On $\mathbb{P}^1$ consider the trivial bundle $\mathcal{O}\oplus \mathcal{O}$, and the subbundle $\mathcal{L}\_{a,b}\subset\mathcal{O}\oplus \mathcal{O}$ that on an open subset $U$ of $\mathbb{P}^1$ is generated by the local sections $(f,g)$ of $\mathcal{O}\oplus \mathcal{O}$ such that $af(p)+bg(p)=0$, where $p\in\mathbb{P}^1$ is a point and $a,b$ are scalars. How can we write $\mathcal{L}\_{a,b}$?


For instance if $a =1, b=0$ then the condition is just $f(p)=0$ and hence $\mathcal{L}\_{1,0} = \mathcal{O}(-1)\oplus\mathcal{O}$. If $a=0, b=1$ the condition is $g(p)=0$ and hence $\mathcal{L}\_{0,1} = \mathcal{O}\oplus\mathcal{O}(-1)$. But what is for instance $\mathcal{L}\_{1,1}$?



Subbundle generated by linearly dependent sections
$\mathcal{L}\_{1,1}$ is still $\mathcal{O}(-1) \oplus \mathcal{O}$. Actually, this is the only subsheaf in $\mathcal{O} \oplus \mathcal{O}$ of rank 2 and degree $-1$.


