The maximal rank conjecture for general algebraic curves

Let $g$, $r$, $d$, and $m$ be positive integers with $r \geq 3$, satisfying

$$g > (r+1)(g-d+r).$$

Let $X \subset \mathbb{P}^r$ be a general curve of genus $g$ and degree $d$. Consider the multiplication map

$$\mu_m: \operatorname{Sym}^m H^0(X, \mathcal{O}_X(1)) \longrightarrow H^0(X, \mathcal{O}_X(m)).$$

The maximal rank conjecture. The map $\mu_m$ is either injective or surjective.

This conjecture predicts the Hilbert function of a general embedded algebraic curve by requiring the natural multiplication maps on global sections to have maximal rank. The paper develops tropical and inductive methods toward it and proves various cases, including the canonical divisor and a wide range of cases for $m=3$; the general statement remains open.