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

TL;DR

This paper studies the one-dimensional boundary strata of the moduli stack of stable curves, relating them to moduli stacks of pointed stable curves and constructing their components as quotient stacks.

Contribution

It introduces a new approach to constructing the one-dimensional boundary components of the moduli stack as quotient stacks, linking them to pointed stable curves.

Findings

Identifies the one-dimensional components as quotient stacks.

Relates boundary components to moduli stacks of pointed stable curves.

Provides a new construction method for these components.

Abstract

The moduli stack of Deligne-Mumford stable curves of genus g admits a stratification, so that the number of nodes of the curves belonging to one stratum is constant. The irreducible components of the stratum corresponding to curves with exactly 3g-4 nodes are one-dimensional substacks. We show how they can be related to moduli stacks of (permutation classes of) pointed stable curves. Using this, we construct all components of this stratum in a new way as quotient stacks.