Wall crossing for derived categories of moduli spaces of sheaves on rational surfaces

TL;DR

This paper advances the understanding of derived categories of moduli spaces of sheaves on rational surfaces by removing a key assumption and establishing semi-orthogonal decompositions related to wall-crossing phenomena.

Contribution

It removes the global quotient presentation requirement in the theory of windows for derived categories of smooth Artin stacks, enabling new semi-orthogonal decompositions for moduli stacks.

Findings

Established semi-orthogonal decompositions for moduli stacks during wall-crossing.

Connected derived categories of moduli spaces with products of Hilbert schemes.

Extended existing results on flipping of strata for Gieseker stability.

Abstract

We remove the global quotient presentation input in the theory of windows in derived categories of smooth Artin stacks of finite type. As an application, we use existing results on flipping of strata for wall-crossing of Gieseker semi-stable torsion-free sheaves of rank two on rational surfaces to produce semi-orthogonal decompositions relating the different moduli stacks. The complementary pieces of these semi-orthogonal decompositions are derived categories of products of Hilbert schemes of points on the surface.