Enumerating Hassett’s Wall and Chamber Decomposition of the Moduli Space of Weighted Stable Curves

ABSTRACT

Hassett constructed a class of modular compactifications of ${\mathcal{M}}_{g,n}$ by adding weights to the marked points. This leads to a natural wall and chamber decomposition of the domain of admissible weights ${\mathcal{D}}_{g,n}$, where the moduli space and universal family remain constant inside a chamber, and may change upon crossing a wall. The goal of this paper is to count the number of chambers in this decomposition. We relate these chambers to a class of boolean functions known as linear threshold functions (LTFs), and discover a subclass of LTFs which are in bijection with the chambers. Using this relation, we prove an asymptotic formula for the number of chambers, and compute the exact number of chambers for n ⩽ 9. In addition, we provide an algorithm for the enumeration of chambers of ${\mathcal{D}}_{g,n}$ and prove results in computational complexity.

Keywords:

• Moduli spaces
• stable curves
• wall-and-chamber
• linear threshold functions
• birational geometry

2000 AMS SUBJECT CLASSIFICATION:

• 14D20
• 14H10
• 05A15
• 03D15

Acknowledgments

We thank Dori Bejleri, Patricio Gallardo, Brendan Hassett, Dave Jensen, Steffen Marcus, Sam Payne, and Dhruv Ranganathan for helpful discussions and suggestions. We thank Nicolle Gruzling for providing us with code used to enumerate linear threshold functions for n ⩽ 9. Finally, we thank the referee for many usual suggestions that helped improve this paper.

Additional information

Funding

This research was completed as part of the SUMRY (Summer Undergraduate Mathematics Research at Yale) program during the summer of 2017, where the first author was a mentor and the other authors were participants. SUMRY is supported in part by NSF Grant CAREER DMS-1149054. K.A. was supported in part by an NSF Postdoctoral Fellowship.