Enumerating Hassett's wall and chamber decomposition of the moduli space of weighted stable curves

TL;DR

This paper counts the chambers in Hassett's wall and chamber decomposition of moduli spaces of weighted stable curves, relating them to linear threshold functions, providing asymptotic formulas, exact counts for small cases, and an enumeration algorithm.

Contribution

It introduces a novel connection between chambers and linear threshold functions, deriving asymptotic formulas and computational methods for counting chambers.

Findings

Derived an asymptotic formula for the number of chambers.

Computed exact chamber counts for up to 9 points.

Developed an algorithm for chamber enumeration and analyzed its complexity.

Abstract

Hassett constructed a class of modular compactifications of the moduli space of pointed curves by adding weights to the marked points. This leads to a natural wall and chamber decomposition of the domain of admissible weights 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 moduli spaces of weighted stable curves with at most 9 points. In addition, we provide an algorithm for the enumeration of the chambers and prove results in computational complexity.