Betti numbers of Brill–Noether varieties on a general curve

We compute the rational cohomology groups of the smooth Brill–Noether varieties Gr d .C /, parametrizing linear series of degree d and dimension exactly r on a general curve C. As an application, we determine the whole intersection cohomology of the singular Brill–Noether loci W r d .C /, parametrizing complete linear series on C of degree d and dimension at least r.