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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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