Nef cones of some Quot schemes on a smooth projective curve

Abstract

Let $C$ be a smooth projective curve over $\mathbb{C}$. Let $n,d\ge 1$. Let $\mathcal{Q}$ be the Quot scheme parameterizing torsion quotients of the vector bundle ${\mathcal{O}}_C^n$ 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 $n\ge d$ 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.

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