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 $\underline{w}$-stability of $\mathcal{O}_C$ and the goodness of $\underline{w}$. 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.