Let G be a finitely generated Fuchsian group of type (p',0)p ' ≥2. Let  1 be a domain. We construct a holomorphic family given isomorphisms   over D, so that for each λ ∈ D there is a complex disc over λ on which X_λ (G) acts invariantly. We prove that (i) if  , there is no holomorphic family of G over D isomorphic to any Bers fiber space; (ii) if  , the holomorphic family is isomorphic to a Bers fiber space π : F(Γ )→ T(Γ ) if and only if G and Γ have the same signature; and (iii) if  , the holomorphic family is isomorphic to a Bers fiber space if and only if the family is identified with a subspace of a Bers fiber space π :F(G)→ T(G) that is determined by a hyperelliptic locus in T(G). The result can be used to study the irreducibility property of a Bers fiber space.