The Explicit Construction of Orders on Surfaces

The study of orders over surfaces is an integral aspect of noncommutative algebraic geometry. Although there is a substantial amount known about orders, relatively few concrete examples have been constructed explicitly. Of those already constructed, most are del Pezzo orders, noncommutative analogues of del Pezzo surfaces, the simplest case. We reintroduce a noncommutative analogue of the well-known commutative cyclic covering trick and implement it to explicitly construct a vast collection of numerically Calabi-Yau orders, noncommutative analogues of surfaces of Kodaira dimension 0. This trick allows us to read off immediately such interesting geometric properties of the order as ramification data and a maximal commutative quotient scheme. We construct maximal orders, noncommutative analogues of normal schemes, on rational surfaces and ruled surfaces. We also use Ogg-Shafarevich theory to construct Azumaya algebras and, more generally, maximal orders on elliptically fibred surfaces. Originality Statement ‘I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or substantial proportions of material which have been accepted for the award of any other degree or diploma at UNSW or any other educational institution, except where due acknowledgement is made in the thesis. Any contribution made to the research by others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis. I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project’s design and conception or in style, presentation and linguistic expression is acknowledged.’