THE DIVISOR CLASSES OF THE SURFACE Z('P('N)) = G(X,Y) OVER FIELDS OF CHARACTERISTIC P > 0
Abstract
This thesis is a study of the divisor class group of normal affine surfaces F (L-HOOK) (,k)('3) defined by equations of the form z('p) = G(x,y), where the ground field k is assumed to be algebraically closed of characteristic p > 0. Surfaces of this type are briefly considered by O. Zariski in {ZA}. P. Blass, who introduced me to this project, investigates their geometry in {BL 1}. P. Samuel, in his 1964 Tata notes ({SA}), describes the class group of several of these surfaces, such as z('p) = xy and z('p) = x('i) + y('j). These notes form the foundation of this work and a discussion of these appears in chapter one. In chapter two, facts concerning the structure of the class group are proved in conjunction with "Ganong's formula" (2.6), which is used extensively in this exploration. The p = 2 case is considered in chapter three. A description of the class group of the surface z('2) = G(,4) (G(,n) is notation for a polynomial of degree n) is given entirely in terms of the coefficients of G(,4). Several questions are posed (3.5) that are further investigated in chapter four where the surface z('3) = G(,4) is studied. A conjecture of M. Artin is mentioned that is confirmed against the surfaces z('2) = G(,6) and z('5) = G(,3) in chapter five and z('3) = G(,6) and z('p) = G(,3), p > 3, in chapter six. Several computations appear in chapter seven, among them, the calculation of Cl(F) for F defined by the equation z('p) = x('p+1) z('p) = x('p+1) + y('p+1) + xy. An isomorphic form of this surface is studied by Zariski in {ZA}. We end this chapter by answering a question asked by Samuel in {SA 2}. In chapter eight we demonstrate that the separability condition in P. Russell's "Affine Theorem of Castelnuovo" ({RU}) is essential by providing an example of a smooth, nonrational surface with factorial coordinate ring. We extend our research in chapters nine and ten to include hypersurfaces defined by equations of the form z('p('m)) = G(x(,l),...,x(,n)). K. Baba in {BA} uses higher order derivations to study their class groups. We develop an alternate, inductive method of attacking Cl(P:z('p('m)) = G(x(,l),...,x(,n))). Many results from earlier chapters are shown to generalize. We complete chapter ten with a description of Cl(A) for Krull domains A such that (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)
Degree
Ph.D.
Subject Area
Mathematics
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