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 Z S 3 and Z 2 meeting in a "degenerate" point {0}, Z 2 − {0} ≃ B GL 2 , while (Z S 3 − {0}), which contains B S 3 as open substack, is a smooth and universally closed algebraic stack. More precisely we show thatwhere X is an explicit smooth non degenerate projective surface inside 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".