Line bundles and curves on a del Pezzo order.

Abstract

Orders on surfaces provided a rich source of examples of noncommutative surfaces. The existence of the analogue of the Picard scheme for orders, has previously been established by Haffmann and Stuhler and in fact Chan and Kulkarni had already computed it for an order on the projective plane ramified on a smooth quartic. In this thesis, I continue this line of work, by studying the Picard and Hilbert schemes for an order on the projective plane ramified on a union of two conics. My main result is that, upon carefully selecting the right Chern classes, the Hilbert scheme is a ruled surface over a genus two curve. Furthermore, this genus two curve is, in itself, the Picard scheme of the order.