  Let $C$ be an irreducible smooth projective curve defined over an
algebraically closed field. We prove that the symmetric product ${\rm
Sym}^d(C)$ has the diagonal property for all $d \geq 1$. For any positive
integers $n$ and $r$, let ${\mathcal Q}_{{\mathcal O}^{\oplus n}_C}(nr)$ be the
Quot scheme parametrizing all the torsion quotients of ${\mathcal O}^{\oplus
n}_C$ of degree $nr$. We prove that ${\mathcal Q}_{{\mathcal O}^{\oplus
n}_C}(nr)$ has the weak point property.
