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Abstract
In this thesis, we investigate those properties of an algebraic set that are determined by its partially ordered set of subvarieties. These properties are called the combinatorial properties of the algebraic set. We say that two algebraic sets are combinatorially equivalent (or homeomorphic) if their partially ordered set of subvarieties are isomorphic. In section one of this thesis we will classify, up to combinatorial equivalence, all affine surfaces over the algebraic closure of a finite field.
In section two we discuss the combinatorial structure of projective and quasi-projective surfaces. We show, by example, that the combinatorial structure is more varied than in the affine case. Although the results are inconclusive, we prove that through each point P on a non-singular plane projective curve C over the algebraic closure of a finite field there is another curve D that meets C exactly at P.
In the last section of this paper we discuss the Picard group of an affine ring. In particular, we have found necessary and sufficient conditions for a one-dimensional affine ring over any field to have torsion Picard group. For affine rings of larger dimensions, we have some partial results.
