  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.
