Stable extensions by line bundles on Riemann surfaces

Question

Is there a compact Riemann surface $X$ and a line bundle $L$ of negative degree on $X$, such that for any nontrivial extension $$ 0 \rightarrow L \rightarrow E \rightarrow L^{-1} \rightarrow 0, $$ $E$ is a stable vector bundle on $X$? Any comment and reference is welcome, thank you.

Answer 1

This never happens. Pick a point $p\in X$; the exact sequence $0\rightarrow L^{2}\rightarrow L^{2}(p)\rightarrow \mathbb{C}_p\rightarrow 0$ gives rise to an exact sequence $0\rightarrow \mathbb{C}\xrightarrow{\ \partial \ } H^1(L^2)\longrightarrow H^1(L^2(p))\rightarrow 0$. The class $e:=\partial (1)$ in $H^1(L^2)\cong \operatorname{Ext}^1(L^{-1},L) $ maps to $0$ in $\operatorname{Ext}^1(L^{-1}(-p),L) $, hence it defines a nontrivial extension of $L^{-1}$ by $L$ which becomes trivial when pulled back to $L^{-1}(-p)$. This means that the extension bundle $E$ contains $L^{-1}(-p)$, hence is not stable.

Answer 2

Let $C$ be a (complex irreducible) curve of genus at least two. Then, for every line bundle $L$ on $C$ of negative degree, there exist extensions of $L^{-1}$ by $L$ which are stable. Furthermore, the generic extension of $L^{-1}$ by $L$ is stable if $\operatorname{deg}(L)>-(g-1)/2.$ This is discussed in Chapter 4, Friedman: Algebraic Surfaces and Holomorphic Vector Bundles and the case when $\operatorname{deg}(L)=-1$ is proven in Theorem 10 of the same chapter.