Homological properties of a certain noncommutative Del Pezzo surface

Recently, de Thanhoffer de Volcsey and Van den Bergh showed that Grothendieck groups of "noncommutative Del Pezzo surfaces" with an exceptional sequence of length 4 are isomorphic to one of three types, the third one not coming from a commutative Del Pezzo surface. In this paper, we adapt the theory of noncommutative $\mathbb{P}^1$-bundles as appearing in the work of Van den Bergh and Nyman to produce a sheaf $\mathbb{Z}$-algebra whose associated Proj has an exceptional sequence of length 4 for which the Gram matrix is of this third type. We show that this noncommutative scheme is noetherian and describe its local structure through the use of our generalized preprojective algebras.