Multigraded linear series and recollement

Given a scheme Y equipped with a collection of globally generated vector bundles E1,…,En, we study the universal morphism from Y to a fine moduli space M(E) of cyclic modules over the endomorphism algebra of E:=OY⊕E1⊕⋯⊕En. This generalises the classical morphism to the linear series of a basepoint-free line bundle on a scheme. We describe the image of the morphism and present necessary and sufficient conditions for surjectivity in terms of a recollement of a module category. When the morphism is surjective, this gives a fine moduli space interpretation of the image, and as an application we show that for a small, finite subgroup G⊂GL(2,k), every sub-minimal partial resolution of A2k/G is isomorphic to a fine moduli space M(EC) where EC is a summand of the bundle E defining the reconstruction algebra. We also consider applications to Gorenstein affine threefolds, where Reid's recipe sheds some light on the classes of algebra from which one can reconstruct a given crepant resolution.