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. Volklein 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-Volklein's result also holds true for any n, as proved by Volklein 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 partial derivative + n(2) + 2n where d is the transcendence degree of K over its minimal subfield (if char(K) = 0) or the generating rank of K over Kp (if char(K) = p > 0). Accordingly, both the "if" and the "only if" part of Smith-V & ouml;lklein's result hold true for every n >= 2.