# Multigraded linear series and recollement

Alastair Craw, Yukari Ito, Joseph Karmazyn

Research output: Contribution to journalArticle

1 Citation (Scopus)

### 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 535-565 31 Mathematische Zeitschrift 289 1-2 27 Oct 2017 https://doi.org/10.1007/s00209-017-1965-1 Published - 1 Jun 2018

### Fingerprint

Linear Series
Morphism
Moduli Space
Algebra
Module
Surjectivity
Gorenstein
Endomorphism
Line Bundle
Threefolds
Vector Bundle
Bundle
Isomorphic
Subgroup
Partial
Necessary Conditions
Generalise
Subset
Sufficient Conditions

### Keywords

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

### ASJC Scopus subject areas

• Mathematics(all)

### Cite this

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

In: Mathematische Zeitschrift, Vol. 289, No. 1-2, 01.06.2018, p. 535-565.

Research output: Contribution to journalArticle

Craw, Alastair ; Ito, Yukari ; Karmazyn, Joseph. / Multigraded linear series and recollement. In: Mathematische Zeitschrift. 2018 ; Vol. 289, No. 1-2. pp. 535-565.
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