Moduli spaces of one dimensional sheaves on log surfaces and Hilbert schemes

Let $X$ be a smooth projective rational surface, $D\subset X$ an effective anticanonical curve, $\beta$ a curve class on $X$ and $\mathfrak{d}=\sum w_i{P}_i$ an effective divisor on $D_{\mathrm{s}\mathrm{m}}$. We consider the moduli space ${\mathcal{M}}_{\beta }(X,D,\mathfrak{d})$ of sheaves on $X$ which are direct images of rank-$1$ torsion-free sheaves on integral curves $C$ in $\beta$ such that $C{\mathrm{\mid }}_D=\mathfrak{d}$, and show that each point of ${\mathcal{M}}_{\beta }(X,D,\mathfrak{d})$ is smooth over a point in the product of the Hilbert schemes of surface singularities of types $A_{w_i-1}$. Hence, ${\mathcal{M}}_{\beta }(X,D,\mathfrak{d})$ has symplectic singularities and admits a unique symplectic resolution.