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We continue the discussion of the decorated on-shell diagrammatics for planarN < 4 Supersymmetric Yang-Mills theories started in arXiv:1510.03642. Inparticular, we focus on its relation with the structure of varieties on theGrassmannian. The decoration of the on-shell diagrams, which physically keepstracks of the helicity of the coherent states propagating along their edges,defines new on-shell functions on the Grassmannian and can introduce novelhigher-order singularities, which graphically are reflected into the presenceof helicity loops in the diagrams. These new structures turn out to havesimilar features as in the non-planar case: the related higher-codimensionvarieties are identified by either the vanishing of one (or more) Pluckercoordinates involving at least two non-adjacent columns, or new relations amongPlucker coordinates. A distinctive feature is that the functions living onthese higher-codimenson varieties can be thought of distributionally as havingsupport on derivative delta-functions. After a general discussion, we explorein 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 amongthem.

On-shell diagrams and the geometry of planar N < 4 SYM theories