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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

14C05 Parametrization (Chow and Hilbert schemes)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
Full Text: DOI EuDML
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