  We continue the discussion of the decorated on-shell diagrammatics for planar
N < 4 Supersymmetric Yang-Mills theories started in arXiv:1510.03642. In
particular, we focus on its relation with the structure of varieties on the
Grassmannian. The decoration of the on-shell diagrams, which physically keeps
tracks of the helicity of the coherent states propagating along their edges,
defines new on-shell functions on the Grassmannian and can introduce novel
higher-order singularities, which graphically are reflected into the presence
of helicity loops in the diagrams. These new structures turn out to have
similar features as in the non-planar case: the related higher-codimension
varieties are identified by either the vanishing of one (or more) Plucker
coordinates involving at least two non-adjacent columns, or new relations among
Plucker coordinates. A distinctive feature is that the functions living on
these higher-codimenson varieties can be thought of distributionally as having
support on derivative delta-functions. After a general discussion, we explore
in some detail the structures of the on-shell functions on Gr(2,4) and Gr(3,6)
on which the residue theorem allows to obtain a plethora of identities among
them.
