This is a report on joint work with Bagnarol and Perroni, available at arxiv:1907.00826. For any locally free coherent sheaf on a fixed smooth projective curve, we study the class, in the Grothendieck ring of varieties, of the Quot scheme that parametrizes zero-dimensional quotients of the sheaf. We prove that this class depends only on the rank of the sheaf and on the length of the quotients. As an application, we obtain an explicit formula that expresses it in terms of the symmetric products of the curve.
If time allows, we will discuss further work of Andrea Ricolfi extending the result from smooth curves to arbitrary smooth projective manifolds, and its application to Bagnarol's thesis (2019) on the Hodge motive of genus zero stable maps to Grassmannians.