The minimal model program for arithmetic surfaces enriched by a Brauer class

We examine the noncommutative minimal model program for orders on arithmetic surfaces, or equivalently, arithmetic surfaces enriched by a Brauer class β. When β has prime index p > 5, we show the classical theory extends with analogues of existence of terminal resolutions, Castelnuovo contraction and Zariski factorisation. We also classify β-terminal surfaces and Castelnuovo contractions, and discover new unexpected behaviour.