Let $X, Y, S$ be noetherian schemes, $X$ flat and quasi-projective over $S$, $Y$ projective over $S$.
- Is the hom-functor $T\mapsto\text{Hom}_T(X_T, Y_T)$ representable?
If $X$ is flat and projective, it is, by a standard graph argument to deduce representability from that of the Hilbert functor of $X\times_SY$. If $X$ is proper and flat, then the hom-functor is an algebraic space.
- If so, is the representing object quasi-projective/projective?