On the dimension of the locus of determinantal hypersurfaces

The characteristic polynomial P_A(x_0, . . . , x_r) of an r-tuple A := (A_1, . . ., A_r) of n imes n-matrices is defined as P_A(x_0, . . . , x_r) := det(x_0 I + x_1 A_1 + . . . + x_r A_r). We show that if r 65 3 and A := (A_1, . . . , A_r) is an r-tuple of n imes n-matrices in general position, then up to conjugacy, there are only finitely many r-tuples A' := (A_1', dots, A_r') such that p_A = p_{A'}. Equivalently, the locus of determinantal hypersurfaces of degree n in P^r is irreducible of dimension (r-1)n^2 + 1