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.

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. 


Yes, if $g > 1$, it is Fano because its Picard group (over algebraic closure, and hence over K) is isomorphic to $\mathbb{Z}$. You can find the needed references , for example, in Drezet and Narasimhan, Invent. Math. 97 (1989), no. 1, 53–94. 

