Grassmannian of $k((z))$: Picard Group, Equations and Automorphisms

This paper aims at generalizing some geometric properties of Grassmannians of finite dimensional vector spaces to the case of Grassmannnians of infinite dimensional ones, in particular for that of $k((z))$. It is shown that the Determinant Line Bundle generates its Picard Group and that the Pl\"ucker equations define it as closed subscheme of a infinite projective space. Finally, a characterization of finite dimensional projective spaces in Grassmannians allows us to offer an approach to the study of the automorphism group.