Nef cones of some Quot Schemes

Let C be a smooth projective curve over 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 Q: = Q₂/ ₂ (O^ rC, k, d) be the Quot scheme parametrizing rank k and degree d quotients of the trivial bundle of rank r. Let QL denote the closed subscheme of the Quot scheme parametrizing quotients such that the quotient sheaf has determinant L. It is known that QL is an integral, normal, local complete intersection, locally factorial scheme of Picard rank 2, when d0. In this article we compute the nef cone and canonical divisor of this variety when d0. We show this variety is Fano iff r=2k+1.