Let $X$ be a smooth projective curve of genus $g\geq 2$. Given a rank two, degree $d=0$ vector bundle $\mathcal{F}$ on $X$, we consider the grassmannian of sub-line bundles of $\mathcal{F}$ of degree $-1$: $$ G(\mathcal{F},(1,-1)):=\{\mathcal{L}\subset \mathcal{F}:\mathcal{L}\text{ is a line bundle of degree $-1$}\}. $$ It naturally embeds in the degree $-1$ Picard variety of $X$, $\mathrm{Pic}^{-1}(X)$, sending $\mathcal{L}$ to the corresponding point of $\mathrm{Pic}^{-1}(X)$. If $g=2$ and $\mathcal{F}$ is stable, this is a one-dimensional subvariety of $\mathrm{Pic}^{-1}(X)$ (This is for example mentioned in the first paragraph of Lange, Narasimhan, Maximal subbundles of rank two vector bundles on curves).

Is it known what this subvariety is? It looks like there is a lot of litterature concerning this and theta divisors but I have not been able to find information on the topology of these subvarieties, more precisely on their cohomology.