  Given a scheme $Y$ equipped with a collection of globally generated vector
bundles $E_1, \dots, E_n$, we study the universal morphism from $Y$ to a fine
moduli space $\mathcal{M}(E)$ of cyclic modules over the endomorphism algebra
of $E:=\mathcal{O}_Y\oplus E_1\oplus\cdots \oplus E_n$. 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 $G\subset \text{GL}(2,k)$, every sub-minimal partial resolution
of $\mathbb{A}^2_k/G$ is isomorphic to a fine moduli space $\mathcal{M}(E_C)$
where $E_C$ 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.
