On the geometry of rank two vector bundles and two-theta divisors on a curve

Abstract

This thesis aims at presenting results and remarks concerning the study of subvarieties of the projective space |2Ɵ| associated to a smooth projective curve C of genus at least 3 and its connections to the moduli space SU(_c)(2) of rank 2 semi-stable vector bundles with trivial determinant. In the first part of the thesis, I present a review of Narasimhan and Ramanan's embedding of SU(_C)(2) in |2Ɵ| for non-hyperelliptic curves of genus 3 ([N-R2]). In particular, I clarify some of the points of their construction (2.3.6) and give complete proofs of lemma 5.1 and lemma 5.2 (see 2.3.4 and 2.3.17). Moreover in section 2.3 I show that lemma 5.4 of [N-R2] is false, providing an extensive counterexample (2.4.3).In the second part, I discuss the Abel-Jacobi stratification of |2Ɵ| for non-hyperelliptic curves of genus at least 3 as introduced in [0-P], which generalises classical subvarieties of |2Ɵ| such as the Kummer variety. I show that the top element of these stratifications is always a hypersurface and compute its degree (3.2.5), then I provide insight into the characterisation of the general element of the stratification (§3.3).