Kavli Affiliate: Yukari Ito
| Summary:
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 $mathcalM(E)$ of cyclic modules over the endomorphism algebra of $E:=mathcalO_Yoplus E_1opluscdots oplus E_n$. 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 $Gsubset textGL(2,k)$, every sub-minimal partial resolution of $mathbbA^2_k/G$ is isomorphic to a fine moduli space $mathcalM(E_C)$ 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.
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