The Stratified Strong Maximal Rank Conjecture for quadrics of bounded rank

Fix positive integers $g,r,d$ with $r\geq 3$ and $\rho(g,r,d)\geq 0$, and let $C$ be a general curve of genus $g$. For a linear series $\ell=(L,V)\in G^r_d(C)$, let $\Sigma_k(C,\ell)\subset\mathbb{P}(\operatorname{Sym}^2V)$ be the variety of quadric forms of rank at most $k$ lying in $\mathbb{P}\ker(\nu(\ell))$, where

$$\nu(\ell):\operatorname{Sym}^2V\to H^0(C,L^2)$$

and

$$q(g,r,d,k)=\max\left\{-1,\binom{r+2}{2}-\binom{r-k+2}{2}-(2d+2-g)\right\}.$$

Here $3\leq k\leq r+1$, and dimension $-1$ denotes an empty locus. Stratified Strong Maximal Rank Conjecture. The degeneracy locus

$$Q^r_{d,k}(C)=\left\{\ell\in G^r_d(C)\mid \dim\Sigma_k(C,\ell)>q(g,r,d,k)\right\}$$

has dimension

$$\dim Q^r_{d,k}(C)=\rho(g,r,d)-1-\left|\binom{r+2}{2}-\binom{r-k+2}{2}-(2d+1-g)\right|=:\beta(g,r,d,k).$$

The conjecture refines the Strong Maximal Rank Conjecture by stratifying according to the rank of the quadrics. Under the additional assumption $r-d+g\leq 1$, the source cites a result proving that $\Sigma_k(C,\ell)$ has the expected dimension for general $\ell$; the behavior over the degeneracy locus for all linear series remains the proposed problem.