# Stable pairs with a twist and gluing morphisms for moduli of surfaces

## TL;DR

This paper introduces an orbifold-based approach to defining stable pairs and surfaces, simplifying deformation theory and enabling new gluing morphisms for moduli spaces of surfaces.

## Contribution

It provides an orbifold framework for stable pairs, making deformation theory more straightforward and establishing functorial gluing morphisms for moduli of surfaces.

## Key findings

- Orbifold approach simplifies deformation theory.
- Existence of functorial gluing morphisms.
- Application to deformation theory of surface pairs.

## Abstract

We propose an alternative definition for families of stable pairs $(X,D)$ over a possibly non-reduced base when $D$ is reduced, by replacing $(X,D)$ with an appropriate orbifold pair $(\mathcal X,\mathcal D)$. This definition of a stable family ends up being equivalent to previous ones, but has the advantage of being more amenable to the tools of deformation theory. Moreover, adjunction for $(\mathcal X,\mathcal D)$ holds on the nose; there is no correction term coming from the different. This leads to the existence of functorial gluing morphisms for families of stable surfaces and functorial morphisms from $(n + 1)$ dimensional stable pairs to $n$ dimensional polarized orbispace. As an application, we study the deformation theory of some surface pairs.