On the Quot scheme $\mathrm{Quot}^{l}_{S}(\mathcal{E})$

We study the geometry of the Quot scheme $\mathrm{Quot}^l_{S}(\mathcal{E})$ of length $l$ coherent sheaf quotients of a locally free sheaf $\mathcal{E}$ on a smooth projective surface $\mathrm{S}$. In particular, we investigate the nature of its singularities, its intersection theory, and the cohomology of sheaves on $\mathrm{Quot}^l_{S}(\mathcal{E})$.