Bundle-extension inverse problems over elliptic curves

Alexandru Chirvasitu
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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.
椭圆曲线上的束扩展反问题
我们证明了一系列结果,其大意是:在明显的必要数值和行列式约束条件下,复椭圆曲线上固定束之间的 "大多数 "蜕变都会产生(共)核,而这些核要么可以事先指定,要么可以满足各种刚性约束条件。例子包括(a) 对于不可分解的$\mathcal{E}$和$\mathcal{E'}$,其斜率和阶数严格按此顺序递增的单态空间,其内核是半稳态的,且具有最大刚度(即(b) 对于不可分解的 $\mathcal{K}$、$\mathcal{E}$ 和稳定的 $\mathcal{F}$,其斜率严格按此顺序递增,且等级和行列式满足明显的可加性约束,则其内核与 $\mathcal{F}$ 同构的嵌入 $\mathcal{K}\to \mathcal{E}$的空间是开放致密的;(c) 这些结果的明显镜像;(d) 将不可分性弱化为半可分性+最大刚性的一般化;(e) 说明各种假设必要性的各种例子。
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