Explicit Description of A Certain Destabilizing Wall of Skyscraper Sheaves on Ruled Surfaces

TL;DR

This paper explicitly describes destabilizing walls of skyscraper sheaves on ruled surfaces using glued stability conditions and introduces the concept of gluing perversity to analyze stability.

Contribution

It provides a new explicit description of stability condition walls on ruled surfaces and introduces the notion of gluing perversity for stability analysis.

Findings

Explicit description of gluing stability conditions on ruled surfaces.

Identification of destabilizing walls for skyscraper sheaves.

Use of deformation of stability conditions from the base curve.

Abstract

We give a explicit description of gluing stability conditions on ruled surfaces by introducing gluing perversity. Moreover, we describe a destabilizing wall of skyscraper sheaves on ruled surfaces by deformation of stability conditions glued from $\widetilde{GL^{+}}(2,\mathbb{R})$-translates of the standard stability condition on the base curve.