A simplicial approach to effective divisors in M¯0,n

Abstract

We study the Cox ring and monoid of effective divisor classes of M¯0,n≅Blℙn−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 weights in R∖{0} 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≥7⁠.