A numerical criterion for simultaneous normalization

Abstract

We investigate conditions for simultaneous normalizability of a family of reduced schemes; that is, 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 $\mathbb{C}$-varieties $(X_y){}_{y\in Y}$ with normal parameter space $Y$—algebraic or analytic—admits a simultaneous normalization if and only if the Hilbert polynomial of the integral closure $\stackrel{̲}{{\mathcal{O}}_{X_y}}$ is locally independent of $y$. When the $X_y$ are curves, projectivity is not needed, and the statement reduces to the well-known $\delta$-constant criterion ofTeissier. The proofs are basically algebraic, analytic results being related via standard techniques (Stein compacta and forth) to more abstract algebraic ones