Let $X$ be a compact Riemann surface of genus $g \geq 1$, and let L be a line bundle over $X$ with $-g < \deg L \leq -\frac{1}{2}g$. Can we always find a flat line bundle $J \in \operatorname{Pic}^0(X)$, i.e. $\deg J = 0$, such that $L$ could be realized as a line subbundle of $J \oplus J^{-1}$?

I guess there exists some line bundle $L$ with $\deg L \in (-g, -\frac{1}{2}g]$, such that for any $J \in \operatorname{Pic}^0(X)$, we could **not** find an inclusion $L \hookrightarrow J\oplus J^{-1}$.

I try to prove my guess, but failed. Does anyone have references or suggestions?

*Remark: the following table is a collection of results I have obtained*

Degree of the prescribed line bundle $L$ | Existence of the flat line bundle $J$ |
---|---|

$-\frac{1}{2}g < \deg L < 0$ | For generic $L$, $\not\exists J \in \operatorname{Pic}^0(X)$, such that $L \hookrightarrow J\oplus J^{-1}$ |

$-g < \deg L \leq-\frac{1}{2}g$ | ? |

$-2g < \deg L \leq -g$ | For any $L$, $\exists\, J \in \operatorname{Pic}^0(X)$, such that $L \hookrightarrow J\oplus J^{-1}$ |

$\deg L \leq -2g$ | For any $L$, and any $J \in \operatorname{Pic}^0(X)$, there always exists an inclusion $L \hookrightarrow J\oplus J^{-1}$ |