Bundle-extension inverse problems over elliptic curves

TL;DR

This paper investigates the typical behavior of morphisms between vector bundles over elliptic curves, showing that under certain conditions, the kernels and cokernels meet specific rigidity and stability criteria, with results applicable to various bundle configurations.

Contribution

It establishes that, under natural constraints, most morphisms between fixed bundles on elliptic curves produce kernels and cokernels with desired stability and rigidity properties, extending previous understanding.

Findings

Most morphisms produce semistable cokernels with minimal automorphisms.

Open dense spaces of embeddings with prescribed cokernels exist under certain slope and rank conditions.

Mirror and generalized results broaden applicability to various bundle types.

Abstract

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.