Stability phenomena in Deligne--Mumford compactifications via Morse theory

Abstract

We study the rational homology of the Deligne--Mumford compactification $\overline{\mathcal M}_{g,n}$ of the moduli space of stable curves via a family of Morse functions, namely the $\text{sys}_T$ functions. Exploiting the geometric and Morse properties of $\text{sys}_T$, including the existence of an index gap and additivity of the Morse index upon gluing maps, we reprove that in low degrees the homology of $\overline{\mathcal M}_{g,n}$ is supported entirely on the boundary $\partial \overline{\mathcal M}_{g,n}$, providing a geometric perspective complementary to Harer's classical result on the virtual cohomological dimension. Furthermore, we establish finite generation and stability phenomena for the rational homology across all genera and numbers of marked points. We show that for each degree $k$, a finite set of homology elements generates all $k$-th homology classes via attaching copies of thrice-marked $\mathbb{P}^1$. This result also recovers previously known stability in the number of marked points, such as Tosteson's theorem.