A stratification of B⁴(2, KC) over a general curve

Let C be a smooth curve of genus g ≥ 10 with general moduli. We show that the Brill– Noether locus B 4 ( 2 , K C ) contains irreducible subvarieties B 3 ⊃ B 4 ⊃ · · · ⊃ B n , where each B k has dimension 3 g − 10 − k and B 3 is an irreducible component of the expected dimension the Brill–Noether number ρ = 3 g − 13.