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

The long root geometry $A_{n,\{1,n\}}(\mathbb{K})$ for the special linear group $\mathrm{SL}(n+1,\mathbb{K})$ 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 $\mathbb{K}$, usually regarded as the {\em natural} embedding of $A_{n,\{1,n\}}(\mathbb{K})$. S. Smith and H. V\"{o}lklein (A geometric presentation for the adjoint module of $\mathrm{SL}_3(\mathbb{K})$, {\em J. Algebra}, vol. 127) have proved that the natural embedding of $A_{2,\{1,2\}}(\mathbb{K})$ is relatively universal if and only if $\mathbb{K}$ is either algebraic over its minimal subfield or perfect with positive characteristic. They also give some information on the relatively universal embedding of $A_{2,\{1,2\}}(\mathbb{K})$ which covers the natural one, but that information is not sufficient to exhaustively describe it. The "if" part of Smith-V\"{o}lklein's result also holds true for any $n$, as proved by V\"{o}lklein in his investigation of the adjoint modules of Chevalley groups (H. V\"{o}lklein, On the geometry of the adjoint representation of a Chevalley group, {\em J. Algebra}, vol. 127). In this paper we give an explicit description of the relatively universal embedding of $A_{n,\{1,n\}}(\mathbb{K})$ which covers the natural one. In particular, we prove that this relatively universal embedding has (vector) dimension equal to $\mathfrak{d}+n^2+2n$ where $\mathfrak{d}$ is the transcendence degree of $\mathbb{K}$ over its minimal subfield (if $\mathrm{char}(\mathbb{K}) = 0$) or the generating rank of $\mathbb{K}$ over ${\mathbb K}^p$ (if $\mathrm{char}(\mathbb{K}) = p > 0$). Accordingly, both the "if" and the "only if" part of Smith-V\"{o}lklein's result hold true for every $n \geq 2$.