Algebraic geometry is based on the study of algebraic varieties: the zero set of finitely many polynomials. The study of discrete and computational geometry focuses on geometric properties of objects which can be specified by finitely many pieces of data; this finite data set is usually represented on a computer, for example, a polynomial can be represented by its coefficients. In this dissertation, I study three problems for which there is significant interaction between these two fields.

The first problem is to classify asymptotically pure varieties: those varieties where each invertible sheaf has only a single cohomology class with maximal growth. As a special case, I classify asymptotically pure toric varieties. Toric varieties are represented by a finite amount of geometric and combinatorial data and therefore can be studied with discrete geometric techniques. I show that the only (Weil-)asymptotically pure toric varieties are quotients of products of projective spaces. This provides supporting evidence for a conjecture of Fedor Bogomolov about a characterization of asymptotically pure varieties.

The second problem is to analyze subdivision-based algorithms in computational geometry. Such algorithms are highly adaptive and, therefore, only perform more work near difficult features. I provide a general technique for computing the complexity of such algorithms; this general technique is applicable to many subdivision-based algorithms. As a special case, I analyze the complexity of a well-known, simple, and numerical real root isolation algorithm called EVAL. The complexity analysis is non-trivial and also requires basic height-type bounds from number theory.

The third problem is to provide a completely numerical subdivision-based algorithm to find a polygonal approximation to a curve even in the presence of singularities. This is a challenging problem because most numerical algorithms lack this type of correctness statement. By appealing to classical algebraic geometry, I provide such a correctness statement for a numerical algorithm which is based on a recent algorithm of Plantinga and Vegter. This is the first completely numerical algorithm for curve approximation which correctly meshes near singularities: it is guaranteed to be topologically correct.