A closer look at the stacks of stable pointed curves

TL;DR

This paper provides a more elementary and comprehensive proof that the stabilization functor in the moduli stacks of stable pointed curves is well-defined and works over the integers, extending previous complex-based results.

Contribution

It offers a complete, elementary proof that the stabilization functor is a functor over the integers, broadening the foundational understanding of moduli stacks.

Findings

Stabilization is a functor in the moduli stacks of stable pointed curves.

The proof is valid over the integers, not just over complex numbers.

The contraction functor is the inverse of the stabilization functor.

Abstract

In the theory of the moduli-stacks of n-pointed stable curves, there are two fundamental functors, contraction and stabilization. These functors are constructed in [4], where they are used to show that the various \bar{M_{g,n}}'s are DM-stacks. We give here a more elementary and more complete proof of the fact that stabilization, as it is defined in [4], is a functor. The fact that contraction is a functor inverse to the stabilization-functor we think is satisfactorily treated in [4]. In a very recent publication [1] there is a proof based on the proof in [4], but it is adapted to the case of moduli stacks over the complex numbers. In this note we work in the most general setting, i.e. over the integers.