  Let C be a smooth projective curve over the field of the complex numbers. We
consider Brill-Noether loci over the moduli of maps from C to the Grassmannian
G(m,n) 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.
