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 C in D n associates the number of branch points whose local monodromy lies in the class C.

We study Ruan's cohomological crepant resolution conjecture [41] for orbifolds with transversal ADE singularities. In the An-case, we compute both the Chen–Ruan cohomology ring [Formula: see text] and the quantum corrected cohomology ring H*(Z)(q1,…,qn). The former is achieved in general, the later up to some additional, technical assumptions. We construct an explicit isomorphism between [Formula: see text] and H*(Z)(-1) in the A1-case, verifying Ruan's conjecture. In the An-case, the family H*(Z)(q1,…,qn) is not defined for q1 = ⋯ = qn = -1. This implies that the conjecture should be slightly modified. We propose a new conjecture in the An-case (Conjecture 1.9). Finally, we prove Conjecture 1.9 in the A2-case by constructing an explicit isomorphism.

Abstract. The main purpose of this paper is to 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 locus (in the moduli space) of curves admitting an effective action by the dihedral group D n . This invariant has later been used in [CLP13] where the results of Livingston [Liv85] and of Dunfield and Thurston have been extended to the ramified case.

Dedicated to Ugo Bruzzo on the occasion of his 60-th birthday.Abstract. 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 "building data" on Y which are equivalent to the existence of such covers π : X → Y . 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.

We prove the cohomological crepant resolution conjecture of Ruan for the weighted projective space ℙ(1,3,4,4). To compute the quantum corrected cohomology ring, we combine the results of Coates–Corti–Iritani–Tseng on ℙ(1,1,1,3) and our previous results.