Fulton's conjecture on F-nef divisors on the moduli space of pointed rational curves

Let $\overline{M}_{0,n}$ be the moduli space of stable $n$-pointed rational curves, and let $\delta_S$ denote the class of the boundary divisor indexed by $S$, with $\delta_S=\delta_{S^c}$. A divisor is written as $D=\sum_S a_S\delta_S$. Fulton's conjecture. If

$$a_{I \cup J}+a_{I \cup K}+a_{I \cup L}\ \geqslant a_I+a_J+a_K+a_L$$

for every partition $\{1,2,\ldots,n\}=I\cup J\cup K\cup L$ into four disjoint nonempty subsets, then $D$ has an effective boundary representation

$$D=\sum_S b_S\delta_S,$$

with $b_S\geqslant0$ for all $S$. This is the original divisor-theoretic formulation of Fulton’s conjecture: it asserts that every $F$-nef divisor is an effective combination of boundary divisors. The paper studies equivalent convex-geometric formulations; the general conjecture is presented as open, with special cases such as $n=7$ treated separately.