Diagonal property of the symmetric product of a smooth curve

Abstract: 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.