# Birational geometry of moduli spaces of perverse coherent sheaves on blow-ups

## TL;DR

This paper explores the birational geometry of moduli spaces of perverse coherent sheaves on blow-ups, connecting wall crossing phenomena with the minimal model program and derived category embeddings.

## Contribution

It demonstrates that Nakajima-Yoshioka's blow-up/blow-down diagram realizes the minimal model program and establishes a fully-faithful embedding between derived categories of these moduli spaces.

## Key findings

- The diagram corresponds to the minimal model program.
- Derived categories of moduli spaces are fully faithfully embedded.
- Provides geometric insights into wall crossing phenomena.

## Abstract

In order to study wall crossing formula of Donaldson type invariants on the blown-up plane, Nakajima-Yoshioka constructed a sequence of blow-up/blow-down diagrams connecting the moduli space of torsion free framed sheaves on projective plane, and that on its blow-up. In this paper, we prove that Nakajima-Yoshioka's diagram realizes the minimal model program. Furthermore, we obtain a fully-faithful embedding between the derived categories of these moduli spaces.