Singularity conditions on the class group of Zariski surfaces

Let k be an algebraically closed field of characteristic p≠0 and Xg⊂Ak3 be a normal surface defined by an equation of the form zp=g(x,y). Assume the number of singularities of Xg is the maximum possible, which is very often the case. This paper defines an equivalence relation on the singularities of Xg in terms of the Hessian from which it derives a fundamental decomposition of the group of Weil divisors of the surface. From the decomposition various results are obtained relating the structure of the equivalence classes to that of the class group.