$\overline{\mathcal{S}}$ has at most two possible singularities, which are of type $A_{a_{1}-1}$ and $A_{a_{2}-1}$ respectively, and the projection map $\overline{\mathcal{S}}\rightarrow\mathcal{S}$ is a partial resolution, with exceptional divisor isomorphic to $\mathbb{C}\mathbb{P}^{1}$.