Maximal Brill–Noether Loci via the Gonality Stratification

We study the restriction of Brill–Noether loci to the gonality stratification of the moduli space of curves of fixed genus. As an application, we give new proofs that Brill–Noether loci with ρ=−1 have distinct support, and for fixed r, we give lower bounds on when one direction of the noncontainments of the maximal Brill–Noether loci conjecture hold for Brill–Noether loci of rank r linear systems. Using these techniques, we also show that Brill–Noether loci corresponding to rank 2 linear systems are maximal as soon as g≥28 and prove the maximal Brill–Noether loci conjecture for g=20.