Given a vector bundle [Formula: see text] on a complex reduced curve [Formula: see text] and a subspace [Formula: see text] of [Formula: see text] which generates [Formula: see text], one can consider the kernel of the evaluation map [Formula: see text], i.e. the kernel bundle [Formula: see text] associated to the pair [Formula: see text]. Motivated by a well-known conjecture of Butler about the semistability of [Formula: see text] and by the results obtained by several authors when the ambient space is a smooth curve, we investigate the case of a reducible curve with one node. Unexpectedly, we are able to prove results which goes in the opposite direction with respect to what is known in the smooth case. For example, [Formula: see text] is actually quite never [Formula: see text]-semistable. Conditions which gives the [Formula: see text]-semistability of [Formula: see text] when [Formula: see text] or when [Formula: see text] is a line bundle are then given.

中文翻译：

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关于具有节点的可约曲线上的核束

给定复约简曲线 [公式：参见文本] 上的向量丛 [公式：参见文本] 和生成 [公式：参见文本] 的 [公式：参见文本] 的子空间 [公式：参见文本]，可以考虑评估图的内核[公式：见文本]，即与对[公式：见文本]关联的内核束[公式：见文本]。受 Butler 关于 [公式：见正文] 的半稳定性的一个著名猜想和几位作者在环境空间是平滑曲线时获得的结果的启发，我们研究了具有一个节点的可约曲线的情况。出乎意料的是，我们能够证明与平滑情况下已知的结果相反的结果。例如，[公式：见文本]实际上绝不是 [公式：见文本]-半稳定的。给出[公式：