The maximal rank conjecture for general Brill–Noether curves

Let $g,r,d\geq 0$, let $X$ be a general curve of genus $g$, and let $({\mathscr L},V)$ be a general linear series of rank $r$ and degree $d$ on $X$ with $\rho=g-(r+1)(r+g-d)\geq 0$. Assume that the series embeds $X$ as a nondegenerate curve in $\mathbb P^r$, and for $m\geq 2$ consider the restriction map

$$\Gamma(\mathbb P^r,\mathscr O(m))\longrightarrow \Gamma(X,\mathscr L^{\otimes m}).$$

Maximal Rank Conjecture. The rank of this restriction map is

$$\min\left\{\binom{r+m}{m},md+1-g\right\},$$

so the dimension of its kernel, equivalently the dimension of the degree-$m$ homogeneous polynomials vanishing on the image of $X$, is

$$\max\left\{0,\binom{r+m}{m}-(md+1-g)\right\}.$$

This conjecture predicts that the restriction map has maximal possible rank, or equivalently that its kernel has minimal possible dimension. It gives the expected dimensions of the equations of a general Brill–Noether curve in every degree, extending the basic question about the defining equations of such curves.