  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.
