On Quot schemes of rank zero quotients over a curve

I will discuss various aspects of the geometry of Quot schemes of torsion quotients on a smooth projective curve. I will describe in particular results on the cohomology of tautological vector bundles over the Quot scheme, which mirror a parallel picture in the geometry of the Hilbert scheme of points over a surface. The subject has an interesting combinatorial and computational flavor. The talk is partly based on joint work with D. Oprea and S. Sam.