Nef and Effective cones of some Quot Schemes

TL;DR

This paper computes the nef cone, effective cone, and canonical divisor of certain Quot schemes over complex curves, establishing conditions under which these schemes are Fano varieties.

Contribution

It provides explicit descriptions of the cones and canonical divisor for Quot schemes of high degree, extending understanding of their geometric properties.

Findings

The nef cone and effective cone are explicitly computed for large degree.

The Quot scheme is Fano if and only if r=2k+1.

The scheme is integral, normal, and locally factorial with Picard rank 2.

Abstract

Let $C$ be a smooth projective curve over $\mathbb{C}$ of genus $g(C)\geqslant 3$ (respectively, $g(C)=2$). Fix integers $r,k$ such that $2\leqslant k\leqslant r-2$, (respectively, $3\leqslant k\leqslant r-2$). Let $\mathcal{Q}:={\rm Quot}_{C/\mathbb{C}}(\mathcal{O}^{\oplus r}_C, k, d)$ be the Quot scheme parametrizing rank $k$ and degree $d$ quotients of the trivial bundle of rank $r$. Let $\mathcal{Q}_L$ denote the closed subscheme of the Quot scheme parametrizing quotients such that the quotient sheaf has determinant $L$. It is known that $\mathcal{Q}_L$ is an integral, normal, local complete intersection, locally factorial scheme of Picard rank 2, when $d\gg0$. In this article we compute the nef cone, effective cone and canonical divisor of this variety when $d\gg0$. We show this variety is Fano iff $r=2k+1$.