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

Diagonal property of the symmetric product of a smooth curve