On the cohomology of Brill–Noether loci over Quot schemes

Let C be a smooth projective irreducible curve over an algebraic closed field k of characteristic 0. We consider Brill–Noether loci over the moduli space of morphisms from C to a Grassmannian G(m,n) of m-planes in kn and the corresponding Quot schemes of quotients of a trivial vector bundle on C compactifying the spaces of morphisms. We study in detail the case in which m=2, n=4. We prove results on the irreducibility and dimension of these Brill–Noether loci and we address explicit formulas for their cohomology classes. We study the existence problem of these spaces which is closely related with the problem of classification of vector bundles over curves.