Nodal curves and polarizations with good properties

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 w-stability of OC and the goodness of 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.

Coherent systems and BGN extensions on nodal reducible curves

“…In [BF20c] the authors proved that good polarizations exist on any stable nodal curve C with p a (C) ≥ 2. If w is good, then O C is w-stable and the converse holds when C is a nodal curve of compact type (see [BF20c, Theorem 3.9]).…”

“…The situation became a little bit better if one choses a polarization which is good. This class of polarizations was introduced by the authors in [BF20c] by observing that depth one sheaves on nodal curves equipped with good polarizations reflect a lot of properties that hold for vector bundles on smooth curves. For curves of compact type, good polarizations are exactly those for which O C is w-stable (in general, O C is not even w-semistable!).…”