The adjoint variety of SL\(_{m+1}\mathbb C\) is rigid to order three (Q2519055)

Using Cartan's method of moving frames the author proves that the adjoint variety of \(Sl_{m+1}{\mathbb C}\) in \({\mathbb P}(\mathfrak{sl}_{m+1}{\mathbb C})\) is rigid to order \(3\). More precisely (Thm 4.6): If \(Y^{2m-1}\subset{\mathbb P}(\mathfrak{sl}_{m+1})={\mathbb P}^{m^2+2m-1}\), \(m>1\), is an algebraic variety so that there exists a framing over a \(3\)-general point \(y\in Y\) at which the non-vanishing coefficients of the \(2\)nd and \(3\)rd Fubini forms are given by certain expressions then \(Y\) is projectively equivalent to the variety of trace-free rank one \((m+1)\times(m+1)\) matrices.