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Cox rings of algebraic stacks

Andreas Hochenegger, Elena Martinengo, Fabio Tonini

Abstract

We give a proper definition of the multiplicative structure of the following rings: the Cox ring of invertible sheaves on a general algebraic stack; and the Cox ring of rank one reflexive sheaves on a normal and excellent algebraic stack. We show that such Cox rings always exist and establish their (non-)uniqueness in terms of an Ext-group. Moreover, we compare our definition with the classical construction of a Cox ring on a variety. Finally, we give an application to the theory of Mori dream stacks.

Cox rings of algebraic stacks

Abstract

We give a proper definition of the multiplicative structure of the following rings: the Cox ring of invertible sheaves on a general algebraic stack; and the Cox ring of rank one reflexive sheaves on a normal and excellent algebraic stack. We show that such Cox rings always exist and establish their (non-)uniqueness in terms of an Ext-group. Moreover, we compare our definition with the classical construction of a Cox ring on a variety. Finally, we give an application to the theory of Mori dream stacks.

Paper Structure

This paper contains 10 sections, 24 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Families of sheaves, torsors and their Cox rings
  3. Families of sheaves
  4. Generalities on torsors
  5. Families of line bundles as torsors by diagonalisable group schemes
  6. Existence and uniqueness of Cox rings
  7. The Cox ring of invertible sheaves
  8. The Cox ring of reflexive sheaves
  9. Divisorial approach for the Cox ring
  10. Applications to Mori dream stacks

Key Result

Theorem A

Let $\mathcal{X}$ be an algebraic stack. Then a Cox ring of line bundles on $\mathcal{X}$, also called a $\mathop{\mathrm{Pic}}\nolimits$-Cox ring $\mathsf{R}_{\mathop{\mathrm{Pic}}\nolimits}(\mathcal{X})$, exists. Moreover, a $\mathop{\mathrm{Pic}}\nolimits$-Cox ring is unique up to isomorphism if which, in particular, holds in the cases that

Theorems & Definitions (82)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Definition 3
  • Definition 4
  • Definition 5
  • ...and 72 more