On kernel bundles over reducible curves with a node

Abstract

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.