Rationality of the Moduli Space of Vector Bundles over a Smooth Curve

This article contains a survey on the conjectured rationality of the moduli space of vector bundles over a smooth curve. The main result is a new proof of stable rationality, which implies Conjecture 1 (stated below) for a large number of cases. We describe the progress which had been made on this problem by Tyurin and Newstead, and explain why the proof does not work in general. Ballico rejuvenated interest in the argument, he was the rst to prove stable rationality and to recognize its importance. These past results provide the proper historical framework for our contribution to the problem. Because one can read about the details from the original sources 17, 14, 2, 5], the emphasis here is on the central ideas. After introducing the moduli spaces and stating the conjecture, we digress brieey to discuss the various relevant notions of rationality before presenting the new results. To start, x: