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Hilbert functions and Betti numbers in a flat family. (English) Zbl 0591.14007
The aim of this paper is to study Hilbert functions (given by the associated homogeneous coordinate rings) and (algebraic) Betti numbers of the fibers of a flat family of closed subschemes of a fixed projective space. If \(f: X\to Y\) is such a family, then one shows that the Hilbert functions \(y\mapsto H(X_y,n)\) are lower semicontinuous. Moreover, for \(V=\{y\in Y\mid H(X_y,n)\) is maximal for all \(n\},\) one gets that \(V\) is open, non-empty when \(Y\) is assumed to be integral and noetherian. A description of \(V\) and the behaviour of the ideals of the fibers over \(V\) are also given. Concerning Betti numbers \(b_i(X_y)\) one proves that they are upper semicontinuous as \(y\) varies in the above set \(V\). Other results concern relations among the Betti numbers.

MSC:
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14C05 Parametrization (Chow and Hilbert schemes)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
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