A theorem on generalized conjugate nets in projective \(n\)-space (Q2651839)

Let \(M_1, \ldots, M_r\) be points on the \(u^1, \ldots, u^r\)-tangents at the point ẞ(M_0\) of an analytic variety \(V\) of \(r\) dimensions in a linear space \(S_n\). It is supposed that the parametric net is a generalized conjugate net [ this Zbl 0046.397)]. Furthermore. it is assumed that the points \(M_\alpha\) \(\alpha = 1,\ldots, r)\) describe varieties \(V_\alpha\), having the property that the tangent plane of each \(V_\alpha\) at \(M_\alpha\) passes through the points \(V_\beta\). It is proved that in this case the parametric net on \(V_\alpha\) is \((\alpha, \beta)\)-conjugate. The tangent plane of \(V_\alpha\) at \(M_\alpha\) intersects the tangent plane of \(V\) at \(M_0\) in an \((r - 1)\)-plane \(\Pi_\alpha\). The \((\beta, \alpha)\)-Laplace transform of \(V_\alpha\) at \(M_\alpha\) is defined as the point of interesection of the \(u^\beta\)-tangent to \(V_\alpha\) at \(M_\alpha\) with the \((r - 2)\)-plane of intersection of \(\Pi_\alpha\), and a neighbouring \(\Pi_\alpha\) as \(\Pi_\alpha\) moves in the direction of the \(u^\alpha\)-tangent. The coordinates of this point are computed.