Discoveries / algebra

The Cone Theorem for Effective Fourfold Pairs in Characteristic $p > 5$

Extending the minimal model program beyond threefolds in positive characteristic is a standing goal of birational geometry. Assuming the log resolution conjecture for all log pairs birational to $X$, the cone theorem holds for projective log canonical, $\mathbb{Q}$-factorial fourfold pairs $(X, \Delta)$ with $K_X + \Delta \equiv M \ge 0$, over bases of positive and mixed characteristic $p > 5$.