Let $K$ be a field of characteristic zero but not algebraically closed. Let $C$ be a smooth projective curve over $K$. Let $r, d$ be two positive integers that are coprime. Consider the moduli space of stable vector bundles of degree $d$ and rank $r$ over $C$ with fixed determinantal line bundle. Is it fano? If so could someone suggest a reference for this fact.



Is the moduli space of stable vector bundles over a smooth projective curve fano?
Actually Drezet and Narasimhan prove that the canonical class is (-2n) times the positive generator of the Picard group, where n = g.c.d (r,d). Hence the moduli space (with fixed determinant) is Fano.


