Stable Reduction via the Log Canonical Model

TL;DR

This paper extends the stable reduction concept to higher dimensions, proving a conjecture in large characteristic and confirming the properness of certain moduli stacks of stable surfaces.

Contribution

It formulates a higher-dimensional stable reduction conjecture and proves it in large characteristic, assuming standard MMP conjectures, leading to new results on moduli stacks.

Findings

Proved the stable reduction conjecture in large characteristic.

Recovered the Hacon-Kovács theorem on moduli stack properness.

Established conditions depending on characteristic and volume.

Abstract

We formulate a stable reduction conjecture that extends Deligne-Mumford's stable reduction to higher dimensions and provide a simple proof that it holds in large characteristic, assuming two standard conjectures of the Minimal Model Program. As a result, we recover the Hacon-Kov\'acs theorem on the properness of the moduli stack $\overline{\mathscr{M}}_{2,v,k}$ of stable surfaces of volume $v$ defined over $k=\overline{k}$, provided that $\operatorname{char}k>C(v)$, a constant depending only on $v$.