重轻Hassett空间的庞加莱多项式
Poincare Polynomials of Heavy-Light Hassett Spaces
AI总结：
本文研究Hassett的重轻模空间的庞加莱多项式，给出其递推公式、显式公式及有序贝尔数的推广与递推关系，丰富了稳定曲线模空间的组合拓扑研究。
AI中文摘要：
稳定亏格0曲线的Deligne-Mumford空间$\boldsymbol{\bar{M}_{0,n}}$的庞加莱多项式已被Keel、Manin等多位学者广泛研究，这些多项式可通过组合性质的递推公式计算，其指数生成函数满足优美的函数方程与微分方程。本文针对Hassett的重轻模空间$\boldsymbol{\bar{M}_{0,w_{m,n}}}$（含$m$个重标记点、$n-m$个轻标记点）提出若干组合公式，将庞加莱多项式以某集合划分格的默比乌斯函数为基础进行递推表达；当$m=2$时对应Losev-Manin空间，本文通过计数有序集合划分给出$\boldsymbol{\bar{M}_{0,w_{2,n}}}$庞加莱多项式的显式公式，并给出类似$\boldsymbol{\bar{M}_{0,n}}$情形的递推公式，利用Losev-Manin空间分层的几何与拓扑性质证明该递推公式，再借助指数生成函数简化递推关系；最后将其推广至有序贝尔数，并给出该推广形式的递推关系。
英文摘要：
The Poincaré polynomials of the Deligne-Mumford space $\overline{M_{0,n}}$ 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 $\overline{M_{0,w_{m,n}}}$, 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 $\overline{M_{0,w_{2,n}}}$ by counting ordered set partitions; and give a recursive formula for this polynomial similar to that in the setting of $\overline{M_{0,n}}$. 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.
