The irreducible components of the moduli space of dihedral covers of algebraic curves

In this paper we introduce a new invariant for the action of a finite group $G$ on a compact complex curve of genus $g$. With the aid of this invariant we achieve the classification of the components of the moduli space of curves with an effective action by the dihedral group $D_n$. This invariant has been used in the meanwhile by the authors in order to extend the genus stabilization result of Livingston and Dunfield and Thurston to the ramified case. This new version contains an appendix clarifying the correspondence between the above components and the image loci in the moduli space M_g (classifying when two such components have the same image).Comment: 37 pages, final version appearing in 'Groups, Geometry and Dynamics

Dihedral Galois covers of algebraic varieties and the simple cases

In this article we investigate the algebra and geometry of dihedral covers of smooth algebraic varieties. To this aim we first describe the Weil divisors and the Picard group of divisorial sheaves on normal double covers. Then we provide a structure theorem for dihedral covers, that is, given a smooth variety Y, we describe the algebraic \u201cbuilding data\u201d on Y which are equivalent to the existence of such covers \u3c0:X\u2192Y. We introduce then two special very explicit classes of dihedral covers: the simple and the almost simple dihedral covers, and we determine their basic invariants. For the simple dihedral covers we also determine their natural deformations. In the last section we give an application to fundamental groups

Cyclic and Abelian coverings of real varieties

We describe the birational and the biregular theory of cyclic and Abelian coverings between real varieties.Comment: 25 pages, dedicated to Slava (Viatcheslav) Kharlamov on the occasion of his 71-st birthday, final version to appear on Mathematische Annale

Irreducibility of the space of dihedral covers of the projective line of a given numerical type

We show in this paper that the set of irreducible components of the family of Galois coverings of P^1_C with Galois group isomorphic to D_n is in bijection with the set of possible numerical types. In this special case the numerical type is the equivalence class (for automorphisms of D_n) of the function which to each conjugacy class \mathcal{C} in D_n associates the number of branch points whose local monodromy lies in the class \mathcal{C}.Comment: 18 pages, to appear in Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., volume in memory of Giovanni Prod

Genus stabilization for the components of moduli spaces of curves with symmetries

In a previous paper, arXiv:1206.5498, we introduced a new homological invariant \e for the faithful action of a finite group G on an algebraic curve. We show here that the moduli space of curves admitting a faithful action of a finite group G with a fixed homological invariant \e, if the genus g' of the quotient curve is sufficiently large, is irreducible (and non empty iff the class satisfies the condition which we define as 'admissibility'). In the unramified case, a similar result had been proven by Dunfield and Thurston using the classical invariant in the second homology group of G, H_2(G, \ZZ). We achieve our result showing that the stable classes are in bijection with the set of admissible classes \e