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Three Dimensional Jacobian Derivations And Divisor Class Groups
Alkarni, Shalan
Alkarni, Shalan
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Abstract
In 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 $g$.
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Date
2016-05-31
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University of Kansas
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Keywords
Mathematics, Algebra, Algebraic Geometry, Class Groups, Commutative Algebra, Divisors, Group of Logarithmic Derivatives