Nef cones of some Quot schemes on a smooth projective curve

TL;DR

This paper investigates the nef cone of Quot schemes on smooth projective curves, providing explicit descriptions for elliptic curves and bounds in general, linking to symmetric products and establishing nef criteria.

Contribution

It offers a complete description of the nef cone for elliptic curves and bounds for general cases, connecting nef cones of Quot schemes to symmetric products of curves.

Findings

Complete nef cone description for elliptic curves

Bounds for nef cone when $n \,\geq\, d$ and $C$ very general

Necessary and sufficient nef criterion for divisors on Quot schemes

Abstract

Let $C$ be a smooth projective curve over $\mathbb C$. Let $n,d\geq 1$. Let $\mathcal Q$ be the Quot scheme parameterizing torsion quotients of the vector bundle $\mathcal O^n_C$ of degree $d$. In this article we study the nef cone of $\mathcal Q$. We give a complete description of the nef cone in the case of elliptic curves. We compute it in the case when $d=2$ and $C$ very general, in terms of the nef cone of the second symmetric product of $C$. In the case when $n\geq d$ and $C$ very general, we give upper and lower bounds for the Nef cone. In general, we give a necessary and sufficient criterion for a divisor on $\mathcal Q$ to be nef.