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

Abstract

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.