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

Abstract

Hassett constructed a class of modular compactifications of Mg,n by adding weights to the marked points. This leads to a natural wall and chamber decomposition of the domain of admissible weights Dg,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 Dg,n and prove results in computational complexity.Acknowledgements. 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.2. Hassett's chamber decomposition D g,n

Filtrations of moduli spaces of tropical weighted stable curves

“…Both the number of chambers and the number of chambers up to symmetry are finite, as showed in [ADGH17]. For every g, there is a unique maximal chamber, namely the one given by the weight datum 1 (n) .…”