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 ${\mathrm{Sym}}^d(C)$ has the diagonal property for all $d\ge 1$. For any positive integers $n$ and $r$, let ${\mathcal{Q}}_{{\mathcal{O}}_C^{\oplus n}}(nr)$ be the Quot scheme parametrizing all the torsion quotients of ${\mathcal{O}}_C^{\oplus n}$ of degree $nr$. We prove that ${\mathcal{Q}}_{{\mathcal{O}}_C^{\oplus n}}(nr)$ has the weak point property.

Final version