Intersection cohomology of pure sheaf spaces using Kirwan's desingularization

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 Poincaré polynomial of ${\mathbf{M}}_n$ using Kirwan's desingularization method. We obtain the intersection Poincaré polynomial of the moduli space for one-dimensional sheaves on del Pezzo surfaces of degree $\ge 8$ by considering wall-crossings of stable pairs and complexes.