Wall in the stability space of the gluing stability conditions on Hirzebruch surfaces

TL;DR

This paper studies the wall structure of stability conditions on Hirzebruch surfaces, focusing on intersections of geometric chambers with walls and the moduli spaces of semistable objects, using gluing constructions and semiorthogonal decompositions.

Contribution

It provides a detailed analysis of the wall structure and moduli spaces in the stability condition space on Hirzebruch surfaces, employing gluing techniques and semiorthogonal decompositions.

Findings

Identified intersections of geometric chambers with walls.

Determined the moduli space of semistable objects.

Analyzed the wall structure using gluing constructions.

Abstract

This paper investigates the wall structure of the space of stability conditions on Hirzebruch surfaces. Using the gluing construction of \cite{CP} and \cite{Uch} with respect to a fixed semiorthogonal decomposition, we focus on two main objectives: observing the intersection of the geometric chamber $U(S) \subset \mathrm{Stab}(\Sigma_e)$ with the walls of the resulting subspace, and determining the moduli space of $\sigma$-semistable objects on this subspace.