Moduli spaces of complex algebraic curves with isomorphic to [formula omitted] groups of automorphisms

Abstract

In this paper we describe connected components of moduli spaces of pairs ( K, G), where K is a complex algebraic curve and H is a group of automorphisms of K, isomorphic to. Any component is described by the genus g o and a class of equivalence of functions relative t Aut. The problem is reduced to a description of classes of topological equivalence of pair ( G: P), where G: P → P is a group, isomorphic to of autohomeomorphisms of an orientable surface P. By pair ( G: P) we construct a subgroup G p ⊂ G, a symplectic form (.,.) p: G p × G p → Z 2Z and a function S p : G → Z +. We prove that for pairs ( G : P) and ( G′ : P′) an isomorphism G → G′ is generated by a homeomorphism ϑ: P → P′ (that is ( a) = ϑaϑ −1) if and only if ( G p ) = G p′ , ( a), ( b)) p′ = ( a, b) p and S p = S p′ .