Perverse coherent sheaves on blow-ups at codimension 2 loci

TL;DR

This paper investigates the relationship between moduli spaces of stable sheaves on a blow-up of a smooth projective variety and the original variety, revealing a sequence of flip-like transformations and applying this to the birational geometry of the Hilbert scheme of two points.

Contribution

It generalizes Nakajima and Yoshioka's result from surfaces to higher dimensions, establishing connections between moduli spaces via flip-like diagrams.

Findings

Moduli spaces of stable sheaves are connected through flip-like diagrams.

The results extend to higher-dimensional varieties beyond surfaces.

Application to the birational geometry of the Hilbert scheme of two points.

Abstract

Let $f \colon X \to Y$ be the blow-up of a smooth projective variety $Y$ along its codimension two smooth closed subvariety. In this paper, we show that the moduli space of stable sheaves on $X$ and $Y$ are connected by a sequence of flip-like diagrams. The result is a higher dimensional generalization of the result of Nakajima and Yoshioka, which is the case of $\dim Y=2$. As an application of our general result, we study the birational geometry of the Hilbert scheme of two points.