Bundle-extension inverse problems over elliptic curves

We prove a number of results to the general effect that, under obviously necessary numerical and determinant constraints, "most" morphisms between fixed bundles on a complex elliptic curve produce (co)kernels which can either be specified beforehand or else meet various rigidity constraints. Examples include: (a) for indecomposable $\mathcal{E}$ and $\mathcal{E'}$ with slopes and ranks increasing strictly in that order the space of monomorphisms whose cokernel is semistable and maximally rigid (i.e. has minimal-dimensional automorphism group) is open dense; (b) for indecomposable $\mathcal{K}$, $\mathcal{E}$ and stable $\mathcal{F}$ with slopes increasing strictly in that order and ranks and determinants satisfying the obvious additivity constraints the space of embeddings $\mathcal{K}\to \mathcal{E}$ whose cokernel is isomorphic to $\mathcal{F}$ is open dense; (c) the obvious mirror images of these results; (d) generalizations weakening indecomposability to semistability + maximal rigidity; (e) various examples illustrating the necessity of the assorted assumptions.