Construction of Hurwitz Spaces and Application to the Regular Inverse Problem

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}(\mathbb F_{p^m}) 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, \varphi (p^m-1)/2+1\eqiv n/2 (\mod 2), p^m\equiv 3(\mod 4) and n>\max\{\varphi(p^m-1),7\}.