Gluing for stable families of surfaces in mixed characteristic

TL;DR

This paper investigates the normalization and gluing of stable families of slc surfaces over DVRs, establishing key results for moduli space properness and slc adjunction in mixed and positive characteristic.

Contribution

It introduces a gluing technique for stable surface families in mixed characteristic, crucial for moduli space compactification and slc adjunction theory.

Findings

Gluing statement established for mixed characteristic surfaces

Results support properness of moduli space of stable surfaces

Provides foundational work for slc adjunction in characteristic >5

Abstract

We study the normalizations of non-normal stable families of slc surfaces over an excellent DVR. In mixed characteristic, we establish a gluing statement that is relevant for the properness of the moduli space of such surfaces. We also study the fibers of such glued families, in mixed and equi-characteristic, and provide an essential result for slc adjunction in residue characteristic >5.