Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Stack of $S_{3}$-covers   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".

27
The categories of this paper are: comp-gas
1
The categories of this paper are: cs
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
19
The categories of this paper are: eess
23
The categories of this paper are: solv-int
14
The categories of this paper are: math-ph
22
The categories of this paper are: chao-dyn
8
The categories of this paper are: hep-ex
32
The categories of this paper are: atom-ph
36
The categories of this paper are: cmp-lg
28
The categories of this paper are: alg-geom
16
The categories of this paper are: stat
25
The categories of this paper are: patt-sol
15
The categories of this paper are: nucl-ex
13
The categories of this paper are: quant-ph