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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 $N$?

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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 $c_1(E/L_2)=(n_3-2n_1)-(n_2-n_1)$.

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  • $\begingroup$ Very neat solution! $\endgroup$ – Ritwik Feb 26 '15 at 22:19

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