The Strong Maximal Rank Conjecture

Let $g,r,d,k$ satisfy

$$g-d+r\geq0,\qquad 0\leq\rho(g,r,d)<r-2,\qquad k\geq2,$$

and let $C$ be a general curve of genus $g$. For $L\in W^r_dC$, define the multiplication map

$$\phi^k_L:\operatorname{Sym}^k H^0(C,L)\longrightarrow H^0(C,L^{\otimes k}).$$

The Strong Maximal Rank Conjecture. The locus of line bundles $L\in W^r_dC$ for which $\phi^k_L$ is not of maximal rank has dimension

$$\rho(g,r,d)-1-\left|{r+k\choose k}-(dk+1-g)\right|.$$

In particular, $\phi^k_L$ is of maximal rank for every $L\in W^r_dC$ when this quantity is negative.

This strengthens the Maximal Rank Conjecture by predicting the dimension of the exceptional locus of linear series on a general curve. The conjecture remains open in general, although the supplied status evidence says that recent tropical methods prove three cases.