Lang, JeffreyAlkarni, Shalan2016-11-032016-11-032016-05-312016http://dissertations.umi.com/ku:14635https://hdl.handle.net/1808/21802In this thesis, we use P. Samuel's purely inseparable descent methods to investigate the divisor class groups of the intersections of pairs of hypersurfaces of the form $w_1^p=f$, $w_2^p=g$ in affine $5$-space with $f$, $g$ in $A=k[x,y,z]$; $k$ is an algebraically closed field of characteristic $p$ $$ $0$. This corresponds to studying the divisor class group of the kernels of three dimensional Jacobian derivations on $A$ that are regular in codimension one. Our computations focus primarily on pairs where $f$, $g$ are quadratic forms. We find results concerning the order and the type of these groups. We show that the divisor class group is a direct sum of up to three copies of $\mathbb{Z}_p$, is never trivial, and is generated by those hyperplane sections whose forms are factors of linear combinations of $f$ and $g$.60 pagesenCopyright held by the author.MathematicsAlgebraAlgebraic GeometryClass GroupsCommutative AlgebraDivisorsGroup of Logarithmic DerivativesDissertationopenAccess