Let $X$ be a smooth projective curve of genus $g$. Consider the Ext space over $J\_{d\_1}\times J\_{d\_2}$ i.e., the vector space of isomorphism classes of extensions of line bundles of degree $d\_2$ by line bundles of degree $d\_1$. ($J\_i$ denotes the Jacobian of degree $i$).


There is an action of $Hom(L, Q)$ on Ext^1(L,Q), where $(L,Q)\in J\_{d\_1}\times J\_{d\_2}$. Is there a vector bundle over $J\_{d\_1}\times J\_{d\_2}$ whose fibre over any given $(L,Q)$ is $\frac{Ext^1(L,Q)}{Hom(L, Q)}$? Notice that as $(L,Q)$ varies the $dim~Ext^1(L,Q)-dim~Hom(L, Q)$ remains constant.


For simplicity let us assume that $dim~Ext^1(L,Q)-dim~Hom(L, Q)\geq 0$.



About universal Ext space
There is a natural action of $\operatorname{Hom}(L,Q)$ on $\operatorname{Ext}^1(L,Q)$, namely the trivial one. 


There is a natural vector bundle stack over $J\_{d\_1}\times J\_{d\_2}$ whose fiber at every point $(L,Q)$ is the stack quotient $[\operatorname{Ext}^1(L,Q)/\operatorname{Hom}(L,Q)]$, that is a groupoid such that its $\pi\_0$ (i.e., set of isomorphism classes of objects) is $\operatorname{Ext}^1(L,Q)$ and each object has automorphism group canonically isomorphic to $\operatorname{Hom}(L,Q)$.


It can be obtained by applying the (arrow-reversing) equivalence between $D^{[-1,0]}\_{\text coh}$ and abelian cone stacks to the perfect complex which is dual to $R\pi\_\*(L^\vee\otimes Q)$, where $$\pi:X\times J\_{d\_1}\times J\_{d\_2}\to J\_{d\_1}\times J\_{d\_2}$$ is the natural projection.


Details of the equivalence between derived category and abelian cone stacks, including the fact that it maps perfect complexes with perfect amplitude in [-1,0] to vector bundle stacks and conversely, can be found in [Behrend-Fantechi](https://arxiv.org/abs/alg-geom/9601010 "The Intrinsic Normal Cone").


