Diagonal property of the symmetric product of a smooth curve

TL;DR

This paper proves that symmetric products of smooth curves possess the diagonal property and that certain Quot schemes have the weak point property, advancing understanding of their geometric structures.

Contribution

It establishes the diagonal property for symmetric products of smooth curves and the weak point property for specific Quot schemes, providing new insights into their geometric features.

Findings

Symmetric products of smooth curves have the diagonal property.

Certain Quot schemes have the weak point property.

Results hold over algebraically closed fields.

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.