Fully faithful functors, skyscraper sheaves, and birational equivalence

Let X and Y be two smooth projective varieties such that there is a fully faithful exact functor from Dᵇ(Coh(X)) to Dᵇ(Coh(Y)). We show that X and Y are birational equivalent if the functor maps one skyscraper sheaf to a skyscraper sheaf. Further assuming that X and Y are of the same dimension, we show that if X has ample canonical bundle and H⁰(X ,KX)≠ 0, or if X is a K3 surface with Picard number one, then Y is birational to a Fourier--Mukai partner of X.