Poincare Polynomials of Heavy-Light Hassett Spaces

Abstract

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.