Fix integers a and b.We prove that for certain projective varieties V ⊂ Pr (e.g. certain possibly singular complete intersections), there are only finitely many components of the Hilbert scheme parametrizing irreducible, smooth, projective, low codimensional subvarieties X of V such that the linear system |aK-bH| is empty. In particular, except for finitely many families of varieties, the canonical map of any irreducible, smooth, projective, low codimensional subvariety X of V, is birational.
