Polytopes of Effective Boundary Expressions of Divisors on ${\bar{M}}_{0,n}$

For a divisor on ${\bar{M}}_{0,n}$, we introduce the polytope of its effective boundary expressions. We establish structural properties of these polytopes under the forgetful maps of ${\bar{M}}_{0,n}$ forgetting marked points, and give equivalent graph-theoretic descriptions. We compute these polytopes for several families of divisors. For psi-classes and their pullbacks by forgetful maps, we show that the polytopes are unimodular simplices. For the log-canonical class and its modifications by psi-classes, we prove that the nonnegative parts of the corresponding polytopes recover spanning forest polytopes and the subtour elimination (Held–Karp relaxation) polytope of the symmetric traveling salesman problem. As an application, we obtain a Minkowski-like decomposition of the subtour elimination polytope into simplices. Finally, for symmetric level-one ${\mathfrak{s}\mathfrak{l}}_p$ conformal block divisors, we show that the defining inequalities are local Turán bounds and the $0\mathrm{/}1$-points are balanced Turán graphs. Moreover, for $p=2$ and $p=n\mathrm{/}2$, these polytopes recover the perfect matching and fractional perfect matching polytopes.