We prove that for certain projective varieties (e.g. smooth complete intersections in the projective space), there are only finitely many components of the Hilbert scheme parametrizing irreducible, smooth, projective subvarieties of low codimension and not of general type. We give similar results concerning subvarieties with globally generated tangent bundle.
Ciliberto, C., DI GENNARO, V. (2004). Boundedness for low codimensional subvarieties.. In V. Popov (a cura di), Algebraic Transformation Groups and Algebraic Varieties (pp. 9-14). Springer.