Abstract
We construct explicitly a finite cover of the moduli stack of compact Riemann surfaces with a given group of symmetries by a smooth quasi-projective variety.
Highlights
Is injective and its image is the stabilizer of [C, [ f ]] in g, that we denote by Stab g ([C, [ f ]])
Let Mg(G) be the stack, in the complex analytic category, whose objects are pairs (π : C → B, α), where π : C → B is a family of compact Riemann surfaces of genus g and α : G ×C → C is an effective action of G on C such that, for any a ∈ G, π ◦ α(a, _) = π
We collect in the following theorem several results about the Teichmüller space, for a proof and for more details we refer to [14,15]
Summary
Throughout the article G is a finite group and g is an integer greater or equal than 2. Let Mg(G) be the stack, in the complex analytic category, whose objects are pairs (π : C → B, α), where π : C → B is a family of compact Riemann surfaces of genus g and α : G ×C → C is an effective (holomorphic) action of G on C such that, for any a ∈ G, π ◦ α(a, _) = π. To the coarse moduli space of compact Riemann surfaces of genus g yields an isomorphism Tg/ g ∼= Mg. there is a universal family of Riemann surfaces of genus g with Teichmüller structure η : Xg → Tg. Let α be an effective action of G on C, viewed as an injective group homomorphism α : G → Aut(C).
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