Let $\varphi:A\to B$ be a homomorphism of finite-dimensional algebras over an algebraically closed field and $\varphi^{(c)}:{\rm mod}_B^c\to{\rm mod}_A^c$ the induced morphism of the associated module schemes for any integer $c\geq 1$. We prove that if the induced functor ${\rm mod\,} B\to{\rm mod\,} A$ is hom-controlled then the restriction of $\varphi^{(c)}$ to any connected component of ${\rm mod}_B^c$ is a composition of a smooth morphism followed by an immersion. Some new results on the types of singularities in the orbit closures of module schemes are also proved.

2000 Mathematical Subject Classification: 14B05, 14L30, 16G10.