  We show that the derived categories of symmetric products of a curve are
embedded into the derived categories of the moduli spaces of vector bundles of
large ranks on the curve. It supports a prediction of the existence of a
semiorthogonal decomposition of the derived category of the moduli space,
expected by a motivic computation. As an application, we show that all Jacobian
varieties, symmetric products of curves and all principally polarized abelian
varieties of dimension at most three, are Fano visitors. We also obtain similar
results for motives.
