Maximal Brill-Noether loci via degenerations and double covers

Using limit linear series on chains of curves, we show that closures of certain Brill–Noether loci contain a product of pointed Brill–Noether loci of small codimension. As a result, we obtain new non-containments of Brill–Noether loci, in particular that all dimensionally expected non-containments hold for expected maximal Brill–Noether loci. Using these degenerations, we also give a new proof that Brill–Noether loci with expected codimension $-\rho \le \lceil g\mathrm{/}2\rceil$ have a component of the expected dimension. Additionally, we obtain new non-containments of Brill–Noether loci by considering the locus of the source curves of unramified double covers.