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

Kenneth Ascher, Connor Dubé, Daniel Gershenson, Elaine Hou

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
Pages (from-to)36-53
Number of pages18
JournalExperimental Mathematics
Volume29
Issue number1
DOIs
StatePublished - Mar 2 2020
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Keywords

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

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