The maximal Brill–Noether loci conjecture

Let $\mathcal{M}^{r}_{g,d}$ denote the Brill–Noether locus of smooth genus-$g$ curves possessing a linear series $g^r_d$, and let $\rho(g,r,d)=g-(r+1)(g-d+r)$ be the Brill–Noether number. A locus $\mathcal{M}^{r}_{g,d}$ is expected maximal if $2\leq d\leq g-1$, $\rho(g,r,d)<0$, $\rho(g,r,d+1)\geq 0$, and, when $r\geq 2$, $\rho(g,r-1,d-1)\geq 0$. Maximal Brill–Noether loci conjecture. In every genus $g\geq 3$, the maximal Brill–Noether loci are the expected maximal loci, except when $g=7,8,9$. This conjecture concerns the classification of Brill–Noether loci after excluding the standard containments arising from adding base points, subtracting non-base points, and Serre duality. The exceptional genera $7$, $8$, and $9$ are explicitly excluded from the asserted classification; the supplied text gives no resolution status.