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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 papercalculates the intersection Poincar\'e polynomial of $\mathbf{M}_n$ usingKirwan's desingularization method. We obtain the intersection Poincar\'epolynomial of the moduli space for one-dimensional sheaves on del Pezzosurfaces of degree $\geq 8$ by considering wall-crossings of stable pairs andcomplexes.

Intersection cohomology of pure sheaf spaces using Kirwan's desingularization