Matrices of linear forms and curves

Abstract In order to understand the triple action of PGL n +1 on the projective space of nonzero ( n  + 1) × ( n  + 1) matrices of linear forms on P n , we associate a quadratic rational map ϕ : P n → P n to any such matrix A . The properties of the dynamical system obtained by iteration of ϕ , some of which are of a geometric nature, generate invariants and a canonical form for the orbit of A . We study a family of matrices parametrized by P 1 , whose associated geometry is given by the rational normal curve for each dimension n  = 2, 3, 4. Our analysis involves the osculating flags to the curves; and we calculate the stabilizers of our rational maps and matrices.