The geometry of $\Pi$-invertible sheaves

Abstract

Using the fact that $\Pi$-invertible sheaves can be interpreted as locally free sheaves of modules for the super skew field $\mathbb{D}$, we give a new construction of the $\Pi$-projective superspace $\mathbb{P}^n_{\Pi, B}$ over affine $k$ superschemes $B$, $k$ an algebraically closed field. We characterize morphisms into $\mathbb{P}^n_{\Pi, B}$ and give a new interpretation of the composition of $\Pi$-invertible sheaves in terms of the algebra of $\mathbb{D}$.