Let $X$ be a compact connected Riemann surface and $E \rightarrow X$ a holomorphic vector bundle of rank 2. Then for any holomorphic sub-line bundle $L \subset E$, there exists a holomorphic section $s$ of $\mathbb{P}(E)$ associated to it.The inverse is also true, that is, for any holomorphic section $s \in \Gamma(X,\mathbb{P}(E))$, we can construct a holomorphic sub-line bundle $L \subset E$.


Now, suppose $E$ is projectively flat, which means that $\mathbb{P}(E)$ is a flat $\mathbb{P}^1$ bundle. We fix a projectively flat structure on $E$, i.e. we choose a family of trivializations $\phi\_i: \mathbb{P}(E)|\_{U\_i} \rightarrow U\_i \times \mathbb{P}^1$ such that the transition function $g\_{ij}: U\_i \cap U\_j \rightarrow PGL(2,\mathbb{C})$ is a constant map, where $g\_{ij}(x) = (\phi\_i \circ \phi\_j^{-1})|\_{\{x\} \times \mathbb{P}^1}$.


Under this projectively flat structure $(U\_i, \phi\_i)$ of $\mathbb{P}(E)$, the section $s \in \Gamma(X,\mathbb{P}(E))$ can be re-written as $(U\_i, s\_i)$ such that


1. $s\_i: U\_i \rightarrow \mathbb{P}^1$ is holomorphic;
2. $\phi\_i \circ s|\_{U\_i}(x) = (x, s\_i(x))$ for $x \in U\_i$.


If $s\_i$ is not a constant map, we could assume $U\_i$ is sufficiently small such that $s\_i$ has at most one branch point $p$ in $U\_i$ and $p \not\in U\_i\cap U\_j$ for any $j\neq i$. Then we define a positive divisor $B\_s = \sum\_{x\in X} \nu\_{s\_i}(x)\cdot x$ for some $i$ such that $x \in U\_i$, where $\nu\_{s\_i}(x)$ is the branching order.


In summary, for any given holomorphic triple $L \subset E$ on $X$ with a fixed projectively flat structure $(U\_i,\phi\_i)$ on $E$ such that


* rank($L$)=1, rank($E$)=2.
* The section $s$ corresponding to $L$ is not constant, i.e. $s\_i$ is not constant for some $i$.


My questions are


1. Can we determine the divisor $B\_s$ directly by $L$ and $E$?
2. What can we say about the relation between $\deg L$, $\deg E$ and $\deg B\_s$?


Any comments, suggestion or references are welcome. Thank you!



Special holomorphic triples on Riemann surfaces and branch divisors
See the formula in the proof of Corollary 11.3.1 in [The Monodromy Groups of Schwarzian Equations on Closed Riemann Surfaces](https://www.math.ucdavis.edu/~kapovich/EPR/GaKaMa.pdf):
$$
deg(L)= g-1 +(deg(E) -deg(B\_s))/2.
$$
As for your first question (*can we*?), the answer is *[yes, we can](https://www.youtube.com/watch?v=jjXyqcx-mYY)*: $g-1$ comes from the half-canonical bundle of your Riemann surface. 


See also R.Mandelbaum's papers: 


Branched structures on Riemann surfaces, Trans. A.M.S. 163
(1972), 261-275.


Branched structures and affine and projective bundles on Riemann
surfaces, Trans. A.M.S. 183 (1973), 37-58.


Unstable bundles and branched structures on Riemann surfaces, Math.
Ann. 214 (1975), 49-59. 


