Minimal rational curves on moduli spaces of stable bundles

Abstract

Let C be a smooth projective curve of genus g ≥ 2 and L be a line bundle on C of degree d. Let M := UC(r,L) be the moduli space of stable vector bundles on C of rank r and with the fixed determinant L. Assume that (r, d) = 1, then M is a smooth projective Fano variety with Picard number 1. For any projective curve on M , we can define its degree with respect to the ample anti-canonical line bundle −KM . The first result of this paper determines all rational curves of minimal degree passing through a generic point of M , which answers a question of Jun-Muk Hwang (see Question 1 in [Hw]).