{"title":"Bundle-extension inverse problems over elliptic curves","authors":"Alexandru Chirvasitu","doi":"arxiv-2407.07344","DOIUrl":null,"url":null,"abstract":"We prove a number of results to the general effect that, under obviously\nnecessary numerical and determinant constraints, \"most\" morphisms between fixed\nbundles on a complex elliptic curve produce (co)kernels which can either be\nspecified beforehand or else meet various rigidity constraints. Examples\ninclude: (a) for indecomposable $\\mathcal{E}$ and $\\mathcal{E'}$ with slopes\nand ranks increasing strictly in that order the space of monomorphisms whose\ncokernel is semistable and maximally rigid (i.e. has minimal-dimensional\nautomorphism group) is open dense; (b) for indecomposable $\\mathcal{K}$,\n$\\mathcal{E}$ and stable $\\mathcal{F}$ with slopes increasing strictly in that\norder and ranks and determinants satisfying the obvious additivity constraints\nthe space of embeddings $\\mathcal{K}\\to \\mathcal{E}$ whose cokernel is\nisomorphic to $\\mathcal{F}$ is open dense; (c) the obvious mirror images of\nthese results; (d) generalizations weakening indecomposability to semistability\n+ maximal rigidity; (e) various examples illustrating the necessity of the\nassorted assumptions.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"25 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Category Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.07344","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.