The strong maximal rank conjecture for general curves

Let $C$ be a general curve of genus $g$, and let $G^r_d(C)$ be the Brill–Noether variety of linear series $l=(L,V)$, with $\rho(g,r,d)\geq0$. For each $l$, let

$$\nu_2(l):\operatorname{Sym}^2(V)\longrightarrow H^0(C,L^{\otimes2})$$

be the multiplication map, and define the failure locus

$$\mathfrak{Quad}_{g,d}^r(C)=\{l\in G^r_d(C):\nu_2(l)\text{ is not of maximal rank}\}.$$

The strong maximal rank conjecture. For a general $[C]\in\mathcal{M}_g$, the locus $\mathfrak{Quad}_{g,d}^r(C)$ has the expected determinantal dimension

$$\dim\mathfrak{Quad}_{g,d}^r(C)=\rho(g,r,d)-1-\left|2d+1-g-{r+2\choose2}\right|,$$

with the convention that a negative expected dimension means that the locus is empty. The conjecture refines maximal-rank predictions by prescribing the dimension of the entire non-maximal-rank locus, not only its emptiness.