### Abstract

Original language | English |
---|---|

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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematische Zeitschrift*,

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

**Multigraded linear series and recollement.** / Craw, Alastair; Ito, Yukari; Karmazyn, Joseph.

Research output: Contribution to journal › Article

*Mathematische Zeitschrift*, vol. 289, no. 1-2, pp. 535-565. https://doi.org/10.1007/s00209-017-1965-1

}

TY - JOUR

T1 - Multigraded linear series and recollement

AU - Craw, Alastair

AU - Ito, Yukari

AU - Karmazyn, Joseph

PY - 2018/6/1

Y1 - 2018/6/1

N2 - 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.

AB - 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.

KW - Linear series

KW - Moduli space of quiver representations

KW - Noncommutative crepant resolutions

KW - Special McKay correspondence

UR - http://www.scopus.com/inward/record.url?scp=85032498183&partnerID=8YFLogxK

U2 - 10.1007/s00209-017-1965-1

DO - 10.1007/s00209-017-1965-1

M3 - Article

VL - 289

SP - 535

EP - 565

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 1-2

ER -