2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/130608We investigate conditions for "simultaneous normalizability" of a family of reduced schemes, i.e., the normalization of the total space normalizes, fiber by fiber, each member of the family. The main result (under more general conditions) is that a flat family of reduced equidimensional projective complex varieties X_y with parameter y ranging over a normal space--algebraic or analytic--admits a simultaneous normalization if and only if the Hilbert polynomial of the integral closure of the structure sheaf O_{X_y} is locally independent of y. When the X_y are curves projectivity is not needed, and the statement reduces to the well known δ-constant criterion of Teissier. Proofs are basically algebraic, analytic results being related via standard techniques (Stein compacta, etc.) to more abstract algebraic ones.35 pages, final version. To appear in Duke Math. JAlgebraic Geometry14B05, 32S15text