Three-dimensional Jacobian derivations and divisor class groups

ABSTRACT

We investigate the divisor class group of the kernels of three dimensional Jacobian derivations on A = k[x,y,z] that are regular in codimension one, where k is an algebraically closed field of characteristic p>0. These correspond to intersections in affine 5-space of pairs of hypersurfaces, $w_1^p=f,w_2^p=g$, with f,g in A. Our calculations focus primarily on pairs where f and g are quadratic forms. We show that in this case the class group is a direct sum of up to three copies of ℤ_p, is never trivial, and is generated by those hyperplane sections whose forms are factors of linear combinations of f and g.

KEYWORDS:

• Divisor class group
• generators
• Jacobian derivation
• kernel
• positive characteristic

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 13A35
• 13N15