A non-commutative deformation of a quadric surface is usually described by a three-dimensional cubic Artin-Schelter regular algebra. In this paper we show that for such an algebra its bounded derived category embeds into the bounded derived category of a commutative deformation of the Hilbert scheme of two points on the quadric. This is the second example in support of a conjecture by Orlov. Based on this example we formulate an infinitesimal version of the conjecture, and provide some evidence in the case of smooth projective surfaces.

A. We introduce graph potentials, which are Laurent polynomials associated to (colored) trivalent graphs. We show that the birational type of the graph potential only depends on the homotopy type of the colored graph, and use this to de ne a topological quantum eld theory. We end by giving an e cient computational method to compute its partition function. This is the rst paper in a series, and we give a survey of the applications of graph potentials in the other parts.

We show that the Poincaré bundle gives a fully faithful embedding from the derived category of a curve of sufficiently high genus into the derived category of its moduli space of bundles of rank r with fixed determinant of degree 1. Moreover we show that a twist of the embedding, together with 2 exceptional line bundles, gives the start of a semi-orthogonal decomposition. This generalises results of Narasimhan and Fonarev-Kuznetsov, who embedded the derived category of a single copy of the curve, for rank 2.

We discuss the projectivity of the moduli space of semistable vector bundles on a curve of genus g ≥ 2. This is a classical result from the 1960s, obtained using geometric invariant theory. We outline a modern approach that combines the recent machinery of good moduli spaces with determinantal line bundle techniques. The crucial step producing an ample line bundle follows an argument by Faltings with improvements by Esteves-Popa. We hope to promote this approach as a blueprint for other projectivity arguments.

We give a description of the Hochschild cohomology for noncommutative planes (resp. quadrics) using the automorphism groups of the elliptic triples (resp. quadruples) that classify the Artin-Schelter regular Z-algebras used to define noncommutative planes and quadrics. For elliptic triples the description of these automorphism groups is due to Bondal-Polishchuk, for elliptic quadruples it is new. 0 14A22, 16E40, 18E30