Linear systems on ruled surfaces and moduli of vector bundles over curves

This work addresses the “moduli problem” for vector bundles over algebraic curves, namely finding a variety which parameterizes the set of vector bundles of rank r and degree d for r > 1. The existence of a moduli space was proved in 1969 by D. Mumford using geometric invariant theory and the technical notion of stability for vector bundles.; When r = 2, there is a geometric presentation of this variety because the moduli problem in this case is related to the classical problem of classifying ruled surfaces. In this work, we exploit the relationships between rank two bundles, ruled surfaces, and affine bundles over a curve to investigate the classification of both rank two bundles and ruled surfaces.; Following work begun in a 1955 paper by M. F. Atiyah, we first classify affine bundles over a curve, and then continue to the classification of ruled surfaces. The relationship between affine bundles, rank two vector bundles, and ruled surfaces is best understood in terms of extensions of line bundles. Each extension corresponds to a section of the corresponding ruled surface.; We show that there is a surjective map to the moduli space of semi-stable bundles from a certain extension space. Each point e of the extension space corresponds to a ruled surface with a fixed section. The base curve can be mapped into this extension space. The ruled surface represented by e has a second section, intersecting the first above the points Q₁,…,Q_r, if and only if e is in the projective space spanned by the images of the points Q₁,…,Q _r.; This determines when two different extensions correspond to the same bundle. It allows us to understand the fibers of the map to the moduli space mentioned above and to classify ruled surfaces. Our work is explicit, and we determine transition matrices for these bundles.; We connect extension spaces to Tyurin parameters, another approach to parameterizing rank two vector bundles. After developing this connection, we suggest an extension of the Tyurin parameters. Finally, we apply the ideas developed to curves of genus 2 and 3.