### Abstract

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.

Original language | English |
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Pages (from-to) | 535-565 |

Number of pages | 31 |

Journal | Mathematische Zeitschrift |

Volume | 289 |

Issue number | 1-2 |

Early online date | 27 Oct 2017 |

DOIs | |

Publication status | Published - 1 Jun 2018 |

### Keywords

- Linear series
- Moduli space of quiver representations
- Noncommutative crepant resolutions
- Special McKay correspondence

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Craw, A., Ito, Y., & Karmazyn, J. (2018). Multigraded linear series and recollement.

*Mathematische Zeitschrift*,*289*(1-2), 535-565. https://doi.org/10.1007/s00209-017-1965-1