locally free over $G^r_d(C)$.

\[strongmaxrank\] We fix integers $g, r, d\geq 1$ as above. For a general $[C]\in
\cM_g$, the locus $$\mathfrak{Quad}_{g, d}^r(C):=\{l\in G^r_d(C): \nu_2(l) \mbox{ is not of maximal rank}\}$$ has the expected dimension as a determinantal variety, that is, $$\mathrm{dim } \ \mathfrak{Quad}_{g, d}^r(C)=\rho(g, r,
d)-1-\bigl|2d+1-g-{r+2\choose 2}\bigr|,$$ where by convention, negative dimension means that $\mathfrak{Quad}_{g, d}^r(C)$ is empty.

The most significant case of Conjecture \[strongmaxrank\] is when we expect that $\mathfrak{Quad}_{g, d}^r(C)=\emptyset$, and we restate $(MRC)_{g, d}^r$ from the Introduction.

\[strongmaxrank2\] We fix integers $g, r, d\geq 1$ such that $$0\leq \rho(g, r, d)< 2d+2-g-{r+2\choose 2}.$$ For a general curve $[C]\in \cM_g$, the map $\nu_2(l)$ is injective for every $l\in G^r_d(C)$.

As discussed in [@AF], various important cases of Conjecture \[strongmaxrank2\] are known, see [@FP], [@F3], [@V1]. We feel that Conjecture \[strongmaxrank2\] should be true, while the evidence for the stronger statement \[strongmaxrank\] is perhaps less compelling and should be regarded more as an open question. It is reassuring to note that Conjecture \[strongmaxrank2\] is compatible with classical Brill-Noether theory.

\[rank4mrc\] $(MRC)_{g, d}^3$ holds. If $d\leq g+1$ and $[C]\in \cM_g$ is a Petri general curve, then $\nu_2(l)$ is injective for every $l\in G^3_d(C)$. Thus $\mathfrak{Quad}_{g, d}^3(C)=\emptyset$.

We fix 