# Multigraded linear series and recollement

## TL;DR

This paper generalizes classical linear series morphisms to multigraded settings, describing their images and conditions for surjectivity using recollement, with applications to resolutions of quotient singularities and Gorenstein threefolds.

## Contribution

It introduces a multigraded linear series framework, characterizes the morphism's image and surjectivity via recollement, and applies these results to resolutions of quotient singularities and Gorenstein threefolds.

## Key findings

- Characterization of the image of the multigraded linear series morphism.
- Necessary and sufficient conditions for surjectivity involving recollement.
- Application to partial resolutions of quotient singularities and Gorenstein threefolds.

## 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.