Balanced complexes and effective divisors on M¯0,n

Abstract

Doran, Giansiracusa, and Jensen showed a bijection between homogeneous elements in the Cox ring of ${\overline{M}}_{0,n}$ not divisible by any exceptional divisor section, and weighted pure-dimensional simplicial complexes satisfying a zero-tension condition. Motivated by the study of the monoid of effective divisors, the pseudoeffective cone and the Cox ring of ${\overline{M}}_{0,n},$ we point out a simplification of the zero-tension condition and study the space of balanced complexes. We give examples of irreducible elements in the monoid of effective divisors of ${\overline{M}}_{0,n}$ for large n. In the case of ${\overline{M}}_{0,7},$ we classify all such irreducible elements arising from nonsingular complexes and give an example of how irreducibility can be shown in the singular case.

Communicated by Lawrence Ein

Keywords:

• Effective divisor
• effective monoid
• moduli space
• simplicial complex

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 14C20
• 14H10
• 05E45
• 14H51

Acknowledgments

We would like to thank Erin Emerson, Connor Halleck-Dubé, Dave Jensen, (Jocelyn) Yuxing Wang, and Nicholas Wawrykow for helpful discussions. We sincerely thank Jeremy Usatine for many insightful conversations. This project started at the program Summer Undergraduate Mathematical Research at Yale (SUMRY). We thank SUMRY and its organizer Sam Payne for their support.