The divisor classes of the hypersurface 𝑧^{𝑝^{𝑚}}=𝐺(𝑥₁,⋯,𝑥_{𝑛}) in characteristic 𝑝>0

Abstract

In this article we use P. Samuel’s purely inseparable descent techniques to study the divisor class groups of normal affine hypersurfaces of the form z p = G ( x 1 , … , x n ) {z^p} = G({x_1},\ldots ,{x_n}) and develop an inductive procedure for studying those of the form z p m = G {z^{p^m}} = G . We obtain results concerning the order and type of these groups and apply this theory to some specific examples.