Aprodu–Farkas strong maximal rank conjecture

Fix integers $g,r,d \geq 1$ such that $0 \leq \rho(g,r,d) < r-2$, and fix $m \geq 2$. Here $\rho(g,r,d)$ is the Brill-Noether number, $C$ is a smooth general curve in $\mathcal{M}_g$, and $G^r_d(C)$ parametrizes linear series $(L,V)$ of degree $d$ and rank $r$ on $C$. The multiplication map is

$$\mu_m\colon \operatorname{Sym}^m V\longrightarrow H^0(C,L^{\otimes m}).$$

Aprodu–Farkas strong maximal rank conjecture. The determinantal variety

$$\{(L,V)\in G^r_d(C)\mid \mu_m\colon \operatorname{Sym}^m V\to H^0(C,L^{\otimes m})\text{ fails to have maximal rank}\}$$

is of expected dimension

$$\rho(g,r,d)-1-\left|{r+m\choose m}-(2d+1-g)\right|,$$

where, by convention, the locus is empty when the expected dimension is negative.

The conjecture concerns the geometry of the locus where a multiplication map fails maximal rank. Its images in the coarse moduli space $\overline{M}_g$ are candidates for interesting effective cycles; the source does not state a resolution.