Intersection cohomology of pure sheaf spaces using Kirwan's desingularization

TL;DR

This paper computes the intersection cohomology of certain moduli spaces of sheaves using Kirwan's desingularization, providing explicit formulas for their intersection Poincaré polynomials and analyzing wall-crossings.

Contribution

It introduces a method to calculate intersection cohomology of pure sheaf spaces via Kirwan's desingularization and applies it to moduli spaces on del Pezzo surfaces.

Findings

Calculated intersection Poincaré polynomial of $ extbf{M}_n$

Derived intersection cohomology for sheaves on del Pezzo surfaces

Analyzed wall-crossings of stable pairs and complexes

Abstract

Let $\mathbf{M}_n$ be the Simpson compactification of twisted ideal sheaves $\mathcal{I}_{L,Q}(1)$ where $Q$ is a rank $4$ quardric hypersurface in $\mathbb{P}^n$ and $L$ is a linear subspace of dimension $n-2$. This paper calculates the intersection Poincar\'e polynomial of $\mathbf{M}_n$ using Kirwan's desingularization method. We obtain the intersection Poincar\'e polynomial of the moduli space for one-dimensional sheaves on del Pezzo surfaces of degree $\geq 8$ by considering wall-crossings of stable pairs and complexes.