Say I know everything about the moduli space $M_G(v)$ of Gieseker (alternatively, slope, Bridgeland, etc.) stable sheaves with Chern character $v$ on a smooth complex projective variety $X$, where $\mathrm{ch}_0 = 1$, i.e. the objects in $M_G(v)$ are rank $1$. Is there a way to extract information about the objects of $M_G(2v)$?
Examples of objects in $M_G(2v)$ would be extensions of objects of the same Hilbert polynomial (slope, phase, etc.) in $M_G(v)$).
I realise that this question is very vague - apologies. If it's too vague, feel free to close. Probably without providing more information, there isn't an easy answer. If anyone's got any sources (or examples) where such a situation arises, I'd be happy to hear/read them. Perhaps there are examples where the extensions mentioned above would be the only objects of $M_G(2v)$ (though this is just a thought; I don't know if such a situation can arise).
Many thanks.