On the properness of the moduli space of stable surfaces over $\mathbb{Z}[1/30]$

TL;DR

This paper proves the properness of the moduli stack of stable surfaces over rac{rac{1}{30}}{}, assuming a conjecture, by establishing a vanishing theorem for certain singularities in characteristic p.

Contribution

It demonstrates properness of the moduli space of stable surfaces over rac{rac{1}{30}}{}, contingent on a local reduction conjecture, using a new vanishing theorem for specific singularities.

Findings

Properness of the moduli stack over rac{rac{1}{30}}{} established.

A local Kawamata--Viehweg vanishing theorem for 3D log canonical singularities proven.

Results depend on the validity of the locally-stable reduction conjecture.

Abstract

We show the properness of the moduli stack of stable surfaces over $\mathbb{Z}[1/30]$, assuming the locally-stable reduction conjecture for stable surfaces. This relies on a local Kawamata--Viehweg vanishing theorem for for 3-dimensional log canonical singularities at closed point of characteristic $p \neq 2, 3$ and $5$ which are not log canonical centres.