Building Data for Stacky Covers

TL;DR

This paper introduces a new framework for understanding stacky covers through stacky building data, establishing an equivalence of categories, and characterizing these covers as flat root stacks with criteria for tamely ramified covers over fields.

Contribution

It defines stacky building data for stacky covers, proves their categorical equivalence, and characterizes stacky covers as flat root stacks, extending previous results to tamely ramified covers.

Findings

Establishes an equivalence between stacky covers and stacky building data.

Shows every stacky cover is a flat root stack.

Provides a criterion for tamely ramified covers over fields.

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.