Nef and Effective cones of some Quot Schemes

Let $C$ be a smooth projective curve over $\mathbb{C}$ of genus $g(C)⩾3$ (respectively, $g(C)=2$). Fix integers $r,k$ such that $2⩽k⩽r-2$, (respectively, $3⩽k⩽r-2$). Let $\mathcal{Q}:={\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}_{C\mathrm{/}\mathbb{C}}({\mathcal{O}}_C^{\oplus r},k,d)$ 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 $d\gg 0$. In this article we compute the nef cone, effective cone and canonical divisor of this variety when $d\gg 0$. We show this variety is Fano iff $r=2k+1$.