Smooth Fano Intrinsic Grassmannians of Type (2, n) with Picard Number Two

We introduce the notion of intrinsic Grassmannians that generalizes the well-known weighted Grassmannians. An intrinsic Grassmannian is a normal projective variety whose Cox ring is defined by the Plucker ideal I_d,n of the Grassmannian Gr(d, n). We give a complete classification of all smooth Fano intrinsic Grassmannians of type (2, n) with Picard number two and prove an explicit formula to compute the total number of such varieties for an arbitrary n. We study their geometry and show that they satisfy Fujita’s freeness conjecture.