Bundle-extension inverse problems over elliptic curves
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917717732229120 |
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| author | Chirvasitu, Alexandru |
| author_facet | Chirvasitu, Alexandru |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07344 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bundle-extension inverse problems over elliptic curves Chirvasitu, Alexandru Algebraic Geometry Category Theory 14H52, 14H60, 14D20, 14F06, 18E10, 18G15, 32L10 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. |
| title | Bundle-extension inverse problems over elliptic curves |
| topic | Algebraic Geometry Category Theory 14H52, 14H60, 14D20, 14F06, 18E10, 18G15, 32L10 |
| url | https://arxiv.org/abs/2407.07344 |