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.

The first author was supported by a Ph.D. Fellowship in Geometry and Mathematical Physics at SISSA. The second and third author were supported by the national projects PRIN 2015EYPTSB-PE1 "Geometria delle varietá algebriche" and 2017SSNZAW 005-PE1 "Moduli Theory and Birational Classification", and by the research group GNSAGA of INDAM. The third author was also supported by FRA 2018 of the University of Trieste.