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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!

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  • $\begingroup$ It was worked out by Mandelbaum in 70s. I will look for references when I have time. $\endgroup$ – Misha Feb 9 '17 at 2:01
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See the formula in the proof of Corollary 11.3.1 in The Monodromy Groups of Schwarzian Equations on Closed Riemann Surfaces: $$ deg(L)= g-1 +(deg(E) -deg(B_s))/2. $$ As for your first question: Yes, we can: $g-1$ 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.

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