The one-dimensional stratum in the boundary of the moduli stack of stable curves

Abstract

It is well-known that the moduli space $\overline{M}_{g}$ of Deligne-Mumford stable curves of genus $g$ admits a stratification by the loci of stable curves with a fixed number $i$ of nodes, where $0 \le i \le 3g-3$. There is an analogous stratification of the associated moduli stack $\overline{\mathcal{M}}_{g}$.

In this paper we are interested in that particular stratum of the moduli stack, which corresponds to stable curves with exactly $3g-4$ nodes. The irreducible components of this stratum are one-dimensional substacks of $\overline{\mathcal{M}}_{g}$. We show how these substacks can be related to simpler moduli stacks of (permutation classes of) pointed stable curves. Furthermore, we use this to construct all of the components of this boundary stratum generically in a new way as explicit quotient stacks.