Nodal curves and polarizations with good properties

In this paper we deal with polarizations on a nodal curve C with smooth components. Our aim is to study and characterize a class of polarizations, which we call “good”, for which depth one sheaves on C reflect some properties that hold for vector bundles on smooth curves. We will concentrate, in particular, on the relation between the w̲-stability of OC and the goodness of w̲. We prove that these two concepts agree when C is of compact type and we conjecture that the same should hold for all nodal curves.