Derived categories of curves as components of Fano varieties

A well known result of Beilinson gives an amazingly simple description of the bounded derived category of coherent sheaves on the projectile line. However, a smooth projective curve C of positive genus does not admit nontrivial semi-orthogonal decompositions. An interesting question is whether such a category can be realised as a semi-orthogonal component of a smooth Fano variety. In particular, it is conjectured that D(C) embeds fully and faithfully into the derived category of the moduli space of rank 2 stable vector bundles on C. We are going to prove this conjecture for a generic curve C. Based on a joint work with Alexander Kuznetsov.