Locating Divisors on Zariski Surfaces

Let k be an algebraically closed field of characteristic p≠0 and X_g⊂A³_k be a normal surface defined by an equation of the form z^p=g(x, y). Let F_g be the field extension of the prime subfield generated by the coefficients of g and assume the number of singularities of X_g is the maximum possible, which is generally the case. This paper describes a finite algebraic extension L_g of F_g such that L_g[x, y, z] contains a basis of the group of Weil divisors of X_g.