Extension maps and the moduli spaces of rank 2 vector bundles over an algebraic curve

Abstract

Let SUc(2,Ʌ) be the moduli space of rank 2 vector bundles with determinant Ʌ on an algebraic curve C. This thesis investigates the properties of a rational map PU(_d,A) →(^c,d) SUc(2, A) where PU(_d,A) is a projective bundle of extensions over the Jacobian J(^d)(C). In doing so the degree of the moduli space SUc(2, Oc) is calculated for non- hyperelliptic curves of genus four (3.4.2). Information about trisecants to the Kummer variety K C SUc(2,Oc) is obtained in sections 4.3 and 4.4. These sections describe the varieties swept out by these trisecants in the fibres of PU1,o(_c) → J(^1)(C) for curves of genus 3, 4 and 5. The fibres of over ϵ(_d) over E ϵ SUc{2,A) are then studied. For certain values of d these correspond to the family of maximal line subbundles of E. These are either zero or one dimensional and a complete description of when these families are smooth is given (5.4.9), (5.4.10). In the one dimensional case its genus is also calculated (if connected) (5.5.5). Finally a correspondence on the curve fibres is shown to exist (5.6.2) and its degree is calculated (5.6.5). This in turn gives some information about multisecants to projective curves (5.7.4), (5.7.7). [brace not closed]