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Building Data for Stacky Covers

Eric Ahlqvist

Abstract

We define stacky building data for stacky covers in the spirit of Pardini and give an equivalence of (2,1)-categories between the category of stacky covers and the category of stacky building data. We show that every stacky cover is a flat root stack in the sense of Olsson and Borne--Vistoli and give an intrinsic description of it as a root stack using stacky building data. When the base scheme S is defined over a field, we give a criterion for when a birational building datum comes from a tamely ramified cover for a finite abelian group scheme, generalizing a result of Biswas--Borne.

Building Data for Stacky Covers

Abstract

We define stacky building data for stacky covers in the spirit of Pardini and give an equivalence of (2,1)-categories between the category of stacky covers and the category of stacky building data. We show that every stacky cover is a flat root stack in the sense of Olsson and Borne--Vistoli and give an intrinsic description of it as a root stack using stacky building data. When the base scheme S is defined over a field, we give a criterion for when a birational building datum comes from a tamely ramified cover for a finite abelian group scheme, generalizing a result of Biswas--Borne.

Paper Structure

This paper contains 10 sections, 47 theorems, 199 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Ramified covers
  3. Deligne--Faltings structures and root stacks
  4. Special Deligne--Faltings data
  5. 2-cocycles in symmetric monoidal categories
  6. Deligne--Faltings data from ramified $D(A)$-covers
  7. Stacky covers as root stacks
  8. Building data for stacky covers
  9. Parabolic sheaves and an application
  10. Closed subgroups of groups of multiplicative type

Key Result

Theorem \ref{thm:main}

Let $\pi\colon \mathscr{X}\to S$ be a stacky cover. Then there exists a canonical (up to canonical isomorphism) building datum$(\mathcal{A},\mathcal{L})$ where $\mathcal{A}=\mathop{\mathrm{Pic}}\nolimits_{\mathscr{X}/S}$ is the relative picard functor, and a canonical isomorphism of stacks where $S_{(\mathcal{A},\mathcal{L})}$ is the root stack associated to the building datum $(\mathcal{A},\math

Figures (1)

  • Figure 1: The red dots represents the elements of $P_A^{int}\subset Q_A^{int}$ and the blue dots represents the elements of $Q_A^{int}\setminus P_A^{int}$ with $A=\mathbb{Z}/3\mathbb{Z}$.

Theorems & Definitions (197)

  • Theorem \ref{thm:main}
  • Theorem \ref{thm:build}
  • Theorem \ref{thm:appl-main}
  • Definition 2.1: Tonini
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 187 more