Let X be a complete variety over an algebraically closed field K, and D an effective (locally principal) divisor. Denote by ND the normal sheaf; it fits into a fundamental exact sequence O+O,-+O,(D)-+N,+O. (1) Consider a flat family .Y of deformations of D; it consists of a parameter variety W, a point w of W, and an effective divisor on X x k W flat over W whose fiber over w is equal to D. Denote by T,(W) the Zariski tangent space to W at w. The characteristic map of Y is a certain linear transfor- mation P: TwP”) - HOW,) and its characteristic system is the linear system on D cut out by the image of p [12, Vol. 1, pp. 94-95; 8, Lect. 22; 7, pp. 479-4801. When W is smooth at w, the completeness of the characteristic system, the surjectivity of p, means, intuitively, that every normal field to D in X can be integrated. Consider the universal flat family & of deformations of D; it is parametrized by the open piece Divtxik) of the Hilbert scheme H = H%/,) . Its characteristic map is always bijective [2, Rem. 5.5, p. 23; 8, Cor. 2, p. 1541. Furthermore, & induces Y by a (unique) map f: W + H. Correspondingly, the two characteristic maps fit into a commutative diagram