Stack of $S_{3}$-covers

TL;DR

This paper investigates the geometric structure of the stack of $S_{3}$-covers, revealing its two irreducible components, their intersection, and explicit descriptions of the smooth component using algebraic geometry techniques.

Contribution

It provides a detailed geometric analysis of the stack of $S_{3}$-covers, including the description of its components and their intersection, and constructs an explicit smooth surface related to these covers.

Findings

The stack has two irreducible components meeting at a degenerate point.

The component containing $ m B S_{3}$ is smooth and can be described as a quotient stack.

An explicit smooth projective surface inside $P^{7}$ is constructed for the $S_{3}$-cover stack.

Abstract

The aim of this paper is to study the geometry of the stack of $S_{3}$-covers. We show that it has two irreducible components $\mathcal{Z}_{S_{3}}$ and $\mathcal{Z}_{2}$ meeting in a "degenerate" point $\{0\}$, $\mathcal{Z}_{2}-\{0\}\simeq \rm B\rm{GL}_{2}$, while $(\mathcal{Z}_{S_{3}}-\{0\})$, which contains $\rm B S_{3}$ as open substack, is a smooth and universally closed algebraic stack. More precisely we show that $\mathcal{Z}_{S_{3}}-\{0\}\simeq[X/\rm{GL}_{2}]$, where $X$ is an explicit smooth non degenerate projective surface inside $\mathbb{P}^{7}$ intersection of five quadrics. All these results are based on the description of certain families of $S_{3}$-covers in terms of "building data".