Let $\mathcal{F}$ be a rank 2 vector bundle over some algebraic variety.
After restrict to the given line $L\cong\mathbb{P}^1$, it fits into an exact sequecne $$ 0 \rightarrow \mathcal{O}_L(-2)\rightarrow \mathcal{F}|L \rightarrow \mathcal{O}_L\oplus\mathcal{O}_q\rightarrow 0, $$ where $q$ denotes a point over $L$. Then for compute the splitting type of $\mathcal{F}|L$,
- The first chern class. Since $c_1(\mathcal{O}_L(-2))=-2$ and $c_1(\mathcal{O}_L\oplus\mathcal{O}_q)=1,$
I have $c_1(\mathcal{F}|L)=-1$ and so I guess $$\mathcal{F}|L\cong\mathcal{O}_L(-2)\oplus\mathcal{O}_L(1)$$ or $$\mathcal{F}|L\cong\mathcal{O}_L(-1)\oplus\mathcal{O}_L.$$
- At that time, I want to show that the one of them is actually cannot fits into above exact sequnce, to conclude what is the splitting type of $\mathcal{F}|L$ is.
Can I get your help? Thank you.