On kernel bundles over reducible curves with a node

Given a vector bundle E on a complex reduced curve C and a subspace V of H^0(E) which generates E, one can consider the kernel of the evaluation map, i.e. the kernel bundle associated to the pair (E,V). Motivated by a well known conjecture of Butler about the semistability of kernel bundles 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, the kernel bundle associated to (E,H^0(E) ) is actually quite never w-semistable. Conditions which gives w-semistability when E is a line bundle are then given.