Families of stable 3-folds in positive characteristic

TL;DR

This paper demonstrates that in positive characteristic, flat families of stable 3-folds do not produce proper moduli spaces, and it constructs examples of log canonical 4-fold pairs with non-weakly normal centers.

Contribution

It reveals fundamental limitations in moduli space construction for stable 3-folds in positive characteristic and provides explicit examples of non-weakly normal log canonical centers.

Findings

Flat families of stable 3-folds are not proper in characteristic p>0

Constructs log canonical 4-fold pairs with non-weakly normal centers

Highlights challenges in moduli theory in positive characteristic

Abstract

We show that flat families of stable 3-folds do not lead to proper moduli spaces in any characteristic $p>0$. As a byproduct, we obtain log canonical 4-fold pairs, whose log canonical centers are not weakly normal.