Stable Reduction via the Log Canonical Model

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-Kovács theorem on the properness of the moduli stack ${\overline{\mathcal{M}}}_{2,v,k}$ of stable surfaces of volume $v$ defined over $k=\overline{k}$, provided that $\mathrm{char}k>C(v)$, a constant depending only on $v$.