Construction of Hurwitz Spaces and Application to the Regular Inverse Problem

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The author give a simple construction of Hurwitz spaces which is defined by Fried and Volklein, and generalize Hurwitz spaces. As a consequence of this construction, the author prove the regularities of the groups PSO+ₙ(Fₚₘ) if p is an odd prime which congruentes with 7 modulo 12, n is an even positive integer grater than 11 and m=1 or p is an odd prime which congruentes with 7 modulo 12, φ(pᵐ-1)/2+1 n/2 (2), pᵐ≡ 3(4) and n>{φ(pᵐ-1),7}.