Noncommutative cyclic covers and maximal orders on surfaces

Abstract

In this paper, we construct various examples of maximal orders on surfaces, including some del Pezzo orders, some ruled orders and some numerically Calabi–Yau orders. The method of construction is a noncommutative version of the cyclic covering trick. These noncommutative cyclic covers are very computable and we give a formula for their ramification data. This often allows us to determine if a maximal order, described via ramification data, can be constructed as a noncommutative cyclic cover. The construction also has applications to Brauer–Severi varieties and, in the quaternion case, we show how to obtain some Brauer–Severi varieties from G-Hilbert schemes of P 1 -bundles.