For every genus g ≤ 12, the paper identifies exactly which Brill–Noether loci are contained in which, yielding a complete relative-position classification.
Maximal Brill--Noether loci via the gonality stratification
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abstract
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 $\rho=-1$ have distinct support, and for fixed $r$ give lower bounds on when one direction of the non-containments 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 $g\ge 28$ and prove the Maximal Brill-Noether Loci Conjecture for $g=20$.
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Brill--Noether loci in genus $\leq 12$
For every genus g ≤ 12, the paper identifies exactly which Brill–Noether loci are contained in which, yielding a complete relative-position classification.