Nodal curves and polarizations with good properties

TL;DR

This paper investigates special polarizations on nodal curves with smooth components, aiming to identify 'good' polarizations that ensure depth one sheaves behave like vector bundles on smooth curves, especially relating stability and polarization properties.

Contribution

It introduces and characterizes 'good' polarizations on nodal curves, establishing their equivalence with stability conditions on compact type curves and conjecturing this for all nodal curves.

Findings

'Good' polarizations are characterized and related to stability.

Equivalence between 'good' polarizations and stability on compact type curves.

Conjecture that this equivalence extends to all nodal curves.

Abstract

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.