Let $E\rightarrow D$ be a complex rank two vector bundle over a compact complex 
one dimensional manifold $D$. Let $L\_1, L\_2 \subset E$ be rank one subbundles of E 
(i.e. line bundles). Let 
$$ n\_1:= \langle c\_1(L\_1), [D]\rangle , ~~n\_2 := \langle c\_1(L\_2), [D]\rangle , ~~n\_3:= \langle c\_1(E), [D]\rangle . $$
If I know the values of $n\_1, n\_2$ and $n\_3$, is this sufficient information 
to calculate the topological intersection number
$$ N:= \mathbb{P}L\_1 \cdot \mathbb{P}L\_2 $$
inside $\mathbb{P}E$, where $\mathbb{P}$ denotes projectivization of a bundle. 
In particular is there a formula for $N$ in terms of $n\_1$, $n\_2$ and $n\_3$?
If not, what further information is required to calculate $N$?



Is there a formula for the intersection of projectivized lines inside a projectivized vector bundle?
It looks like $n\_3-n\_1-n\_2$, but double check the computation. Tensor everything by $L\_1^{-1}$ to make $L\_1$ trivial and recompute the classes to get $0$, $n\_2-n\_1$, and $n\_3-2n\_1$. Then project a section of (trivial now) $L\_1$ to $E/L\_2$: you are interested in the zeroes of this projection, which are counted by $c\_1(E/L\_2)=(n\_3-2n\_1)-(n\_2-n\_1)$.


