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Betti numbers for the Hilbert function strata of the punctual Hilbert scheme in two variables. (English) Zbl 0714.14004
Let k be an algebraically closed field, $$R=k[[x,y]]$$, $$m=(x,y)$$ the maximal ideal of R and $$h(I)(z)=\sum h_ i(I)z^ i$$ the Hilbert function of an ideal I of R of $$colength\quad n$$ where $$h_ i(I)=\dim_ k(m^ i/((I\cap m^ i)+m^{i+1}))$$. For a fixed polynomial h with nonnegative integer coefficients and $$h(1)=n$$ the ideals I with Hilbert function $$h(I)=h$$ are parametrized by a locally closed subscheme $$Z_ h$$ of the punctual Hilbert scheme $$Hilb^ nR$$ and give a stratification $$Hilb^ nR=\cup_{h(1)=n}Z_ h$$ [A. A. Iarrobino, Mem. Am. Math. Soc. 188 (1977; Zbl 0355.14001), Bull. Am. Math. Soc. 78, 819-823 (1972; Zbl 0268.14002) and J. Briançon, Invent. Math. 41, 45-89 (1977; Zbl 0353.14004)].
The author constructs a cellular decomposition of the strata $$Z_ h$$ and computes their Betti numbers by modifying the cellular decomposition of $$Hilb^ n{\mathbb{P}}_ 2$$ given by G. Ellingsrud and S. A. Strømme [Invent. Math. 87, 343-352 (1987; Zbl 0625.14002)].
Reviewer: A.Papantonopoulou

##### MSC:
 14C05 Parametrization (Chow and Hilbert schemes) 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
##### Keywords:
Hilbert stratum; Hilbert function
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##### References:
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