Intrinsic Grassmannians

 

Generalizing the well-known weighted Grassmanians introduced by Corti and Reid, we study certain geometric invariant theory quotients of the affine cone over the Plücker embedding of Gr(k,n). More precisely, we consider normal projective varieties whose Cox ring is defined by the Plücker ideal I_{k,n}. For k = 2, we classify the smooth Fano ones having Picard number two and give a concrete formula to compute their number for arbitrary n.