A simplicial approach to effective divisors in $\overline{M}_{0,n}$

Abstract

We study the Cox ring and monoid of effective divisor classes of $\overline{M}_{0,n} = Bl\mathbb{P}^{n-3}$, over a ring R. We provide a bijection between elements of the Cox ring, not divisible by any exceptional divisor section, and pure-dimensional singular simplicial complexes on {1,...,n-1} with nonzero weights in R satisfying a zero-tension condition. This leads to a combinatorial criterion, satisfied by many triangulations of closed manifolds, for a divisor class to be among the minimal generators for the effective monoid. For classes obtained as the strict transform of quadrics, we present a complete classification of minimal generators, generalizing to all n the well-known Keel-Vermeire classes for n=6. We use this classification to construct new divisors with interesting properties for all n > 6.