The relatively universal cover of the natural embedding of the long root geometry for the group SL ( n + 1 , 𝕂 )

The long root geometry A n , { 1 , n } ( 𝕂 ) for the special linear group SL ( n + 1 , 𝕂 ) admits an embedding in the (projective space of) the vector space of the traceless square matrices of order n + 1 with entries in the field 𝕂 , usually regarded as the natural embedding of A n , { 1 , n } ( 𝕂 ) . S. Smith and H. Völklein in [10] have proved that the natural embedding of A 2 , { 1 , 2 } ( 𝕂 ) is relatively universal if and only if 𝕂 is either algebraic over its minimal subfield or perfect with positive characteristic. They also give some information on the relatively universal embedding of A 2 , { 1 , 2 } ( 𝕂 ) which covers the natural one, but that information is not sufficient to exhaustively describe it. The “if” part of Smith-Völklein’s result also holds true for any n , as proved by Völklein in [13] in his investigation of the adjoint modules of Chevalley groups. In this paper we give an explicit description of the relatively universal embedding of A n , { 1 , n } ( 𝕂 ) which covers the natural one. In particular, we prove that this relatively universal embedding has (vector) dimension equal to 𝔡 + n 2 + 2 n where 𝔡 is the transcendence degree of 𝕂 over its minimal subfield (if char ( 𝕂 ) = 0 ) or the generating rank of 𝕂 over 𝕂 p (if char ( 𝕂 ) = p > 0 ). Accordingly, both the “if” and the “only if” part of Smith-Völklein’s result hold true for every n ≥ 2