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

TL;DR

This paper proves that most isomeric supergrassmannians have trivial Picard and Pi-Picard groups, and provides methods to compute these groups for supertorus orbit closures, highlighting the role of Pi-projective spaces.

Contribution

It offers an elementary proof of triviality of Picard groups for isomeric supergrassmannians and extends techniques to compute Picard groups of supertorus orbit closures from their polytopes.

Findings

Picard and Pi-Picard groups are trivial for most supergrassmannians.

Picard groups of supertorus orbit closures can be computed from their defining polytopes.

Constructs of invertible sheaves depend on Pi-projective space factors.

Abstract

We provide an elementary proof that with the exceptions of certain $\Pi$-projective spaces, both the Picard group and the $\Pi$-Picard set of the isomeric (i.e. type-Q) supergrassmannian are trivial. We extend this technique to show that the Picard group and the $\Pi$-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 $\Pi$-invertible sheaves depends entirely on factors of $\Pi$-projective space, we construct them as symmetric powers of the tautological sheaf and its dual.