A Simplicial Approach to Effective Divisors

We study the Cox ring and monoid of effective divisor classes ofM0,n ∼= BlPn−3, over an arbitrary 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 weights in R\{0} satisfying a zero-tension condition. This leads to a combinatorial criterion for a divisor class to be among the minimal generators for this monoid. Many classical triangulations of closed manifolds yield minimal generators, and we translate some previously studied divisor classes into this simplicial language. 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, for all n ≥ 7, (1) divisor classes whose effectivity depends on R, (2) necessary generators of the Cox ring whose class does not lie on an extremal ray of the effective cone, and (3) counterexamples to the Castravet-Tevelev hypertree conjecture (and for n ≥ 9, new extremal rays of the effective cone). We conclude with several open questions and topics for further investigation.