Jeong-Seop Kim, Stability of Symmetric Powers of Vector Bundles on a Curve

For a stable vector bundle E on a smooth projective curve, it is known that the symmetric powers S^k E are semi-stable and are stable for all k > 0 in sufficiently general. Moreover, if E has rank 2, then S^k E is destabilized by a line subbundle if and only if the ruled surface P_C(E) admits a k-section of zero self-intersection. In this talk, concentrating on the case of rank 2, we will find answers to the questions of which E has strictly semi-stable S^k E, and how many such E there are. Also, we will introduce relations between such E and the orthogonal bundles when k = 2, and Nori’s finite bundles when k ≥ 3.