Invertible sheaves and $\mathrm{\Pi }$-invertible sheaves on the isomeric supergrassmannian and its toric subvarieties

We provide an elementary proof that with the exceptions of certain $\mathrm{\Pi }$-projective spaces, both the Picard group and the $\mathrm{\Pi }$-Picard set of the isomeric (i.e. type-Q) supergrassmannian are trivial. We extend this technique to show that the Picard group and the $\mathrm{\Pi }$-Picard set of a supertorus orbit closure within the isomeric supergrassmannian can be easily calculated from its defining polytope by counting the number of simplex factors. Since the presence of nontrivial invertible sheaves and $\mathrm{\Pi }$-invertible sheaves depends entirely on factors of $\mathrm{\Pi }$-projective space, we construct them as symmetric powers of the tautological sheaf and its dual.