  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 $r=2k+1$.
