Lang's geometric conjecture for varieties of general type

Let $W$ be a variety of general type. Let $\Xi_W \subset W$ denote the locus formed by the union of all irreducible positive-dimensional subvarieties of $W$ that are not of general type. Geometric Lang Conjecture. The subset $\Xi_W$ is a proper closed subvariety of $W$.

This conjecture predicts that the subvarieties obstructing the general-type behavior of $W$ are confined to a proper algebraic subset. It is used in the paper to derive uniform bounds for exceptional loci in families of surfaces of general type and, in particular, for rational and elliptic curves on smooth quintic surfaces.