Stack of $S_3$-covers

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 $\{0\}$, ${\mathcal{Z}}_2-\{0\}\simeq \mathrm{B}{\mathrm{GL}}_2$, while $({\mathcal{Z}}_{S_3}-\{0\})$, which contains $\mathrm{B}{\mathrm{S}}_3$ as open substack, is a smooth and universally closed algebraic stack. More precisely we show that ${\mathcal{Z}}_{S_3}-\{0\}\simeq [X/{\mathrm{GL}}_2]$, 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".

33 pages

References in corpus (1)

• Dihedral Galois covers of algebraic varieties and the simple cases