Let $M$ be a fine moduli space of vector bundles on curve which is an algebraic variety as well. The first example of such an object that I have in mind is rank 2, deg 1 VB on a genus 2 curve. This is an intersection of 2 4-dimensional quadrics, and it is Fano. If I recall correctly, all moduli spaces of bundles with odd degree on an algebraic curve are fine.
My question is: are all fine moduli varieties of VB on an algebraic curve Fano? If not, please give counterexamples.