Poincare Polynomials of Heavy-Light Hassett Spaces
Abstract
The Poincaré polynomials of the Deligne-Mumford space M0,n̅ of stable genus 0 curves have been widely studied by several authors such as Keel and Manin. These polynomials can be computed via a recursive formula that is combinatorial in nature, and their exponential generating functions satisfy elegant functional and differential equations. In this paper, we state some combinatorial formulas to the Poincaré polynomials of Hassett's heavy-light moduli spaces M0,wm,n̅, with m heavy marked points and n−m light marked points. We express the Poincaré polynomials recursively in terms of the Möbius function of a certain lattice of set partitions. In the case of m=2, we get a Losev-Manin space. We give an explicit formula for the Poincaré polynomial or M0,w2,n̅ by counting ordered set partitions; and give a recursive formula for this polynomial similar to that in the setting of M0,n̅. We prove this recursive formula using geometric and topological properties of the stratification of Losev-Manin spaces; and use an exponential generating function to simplify this recursive formula. Finally, we give a remarkable generalization to the ordered Bell numbers and a recurrence relation for this generalization.
Related
- Heavy-Light Wall Crossings and Motivic Asymptotics of Moduli Spaces of Curves. Part I: Stabilization Graphs and Euler Characteristics
- Palindromic Poincaré polynomials in Bn
- Hodge-Deligne and Poincaré polynomials of ℳ̅1,n
- Doubling, Poincaré and Gromov-Hausdorff precompactness for tamed spaces
- A fractional modular Poincaré--Wirtinger-type inequality in Sobolev spaces with variable exponents
BibTeX
@misc{liu2026poincare,
title = {{Poincare Polynomials of Heavy-Light Hassett Spaces}},
author = {Haggai Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03268},
eprint = {2609.03268},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03268},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Poincare Polynomials of Heavy-Light Hassett Spaces
AU - Haggai Liu
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.03268
UR - https://arxiv.org/abs/2609.03268
AB - The Poincaré polynomials of the Deligne-Mumford space M_0,n̅ of stable genus 0 curves have been widely studied by several authors such as Keel and Manin. These polynomials can be computed via a recursive formula that is combinatorial in nature, and their exponential generating functions satisfy elegant functional and differential equations. In this paper, we state some combinatorial formulas to the Poincaré polynomials of Hassett's heavy-light moduli spaces M_0,w_m,n̅, with m heavy marked points and n−m light marked points. We express the Poincaré polynomials recursively in terms of the Möbius function of a certain lattice of set partitions. In the case of m=2, we get a Losev-Manin space. We give an explicit formula for the Poincaré polynomial or M_0,w_2,n̅ by counting ordered set partitions; and give a recursive formula for this polynomial similar to that in the setting of M_0,n̅. We prove this recursive formula using geometric and topological properties of the stratification of Losev-Manin spaces; and use an exponential generating function to simplify this recursive formula. Finally, we give a remarkable generalization to the ordered Bell numbers and a recurrence relation for this generalization.
ER -