  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.
