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

TL;DR

This paper extends the classical minimal model program to arithmetic surfaces with Brauer classes of prime index greater than 5, establishing existence results, classifications, and revealing new phenomena.

Contribution

It develops the noncommutative minimal model program for orders on arithmetic surfaces enriched by Brauer classes, including existence of resolutions and classifications for prime index cases.

Findings

Classical theory extends for prime index > 5

Existence of terminal resolutions and Castelnuovo contractions proven

New unexpected behaviors discovered in the classification

Abstract

We examine the noncommutative minimal model program for orders on arithmetic surfaces, or equivalently, arithmetic surfaces enriched by a Brauer class $\beta$. When $\beta$ 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 $\beta$-terminal surfaces and Castelnuovo contractions, and discover new unexpected behaviour.