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

Hassett constructed a class of modular compactifications of 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 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 D_g,n and prove results in computational complexity.