Line Bundles and Curves on a del Pezzo Order

Orders on surfaces provide a rich source of examples of noncommutative surfaces. In [HS05] the authors prove the existence of the analogue of the Picard scheme for orders and in [CK11] 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. Throughout this paper we assume all objects and maps are defined over an algebraically closed field k of characteristic zero. We denote the dimension of any cohomology group over k by the name of the group written with a non-capital letter for e.g. extA (M,N) := dimkExt i A (M,N) and similarly for h and hom.