Gluing theory for slc surfaces and threefolds in positive characteristic

TL;DR

This paper develops a gluing theory for semi-log canonical (slc) surfaces and threefolds in positive characteristic, enabling advances in understanding their structure and moduli spaces, with specific results for characteristics p>5.

Contribution

It extends Kollár's gluing theory to positive characteristic, covering all p for surfaces and p>5 for threefolds, and introduces new tools for studying nodes and lc centers.

Findings

Established a gluing theory for slc surfaces in all positive characteristics.

Developed a theory of sources and springs for threefolds in characteristic p>5.

Proved projectivity of the moduli space of stable surfaces in characteristic p>5.

Abstract

We develop a gluing theory in the sense of Koll\'{a}r for slc surfaces and threefolds in positive characteristic. For surfaces we are able to deal with every positive characteristic $p$, while for threefolds we assume that $p>5$. Along the way we study nodes in characteristic $2$ and establish a theory of sources and springs \`a la Koll\'{a}r for threefolds. We also give applications to the topology of lc centers on slc threefolds, and to the projectivity of the moduli space of stable surfaces in characteristic $p>5$.