Line bundles and curves on a del Pezzo order

Orders on surfaces provide a rich source of examples of noncommutative surfaces. In Hoffmann and Stuhler (2005) [10] the authors prove the existence of the analogue of the Picard scheme for orders and in Chan and Kulkarni (2011) [7] the Picard scheme is explicitly computed for an order on P2 ramified on a smooth quartic. In this paper, we continue this line of work, by studying the Picard and Hilbert schemes for an order on P2 ramified on a union of two conics. Our 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.