an:04175121
Zbl 0714.14004
G??ttsche, Lothar
Betti numbers for the Hilbert function strata of the punctual Hilbert scheme in two variables
EN
Manuscr. Math. 66, No. 3, 253-259 (1990).
00155382
1990
j
14C05 13D40
Hilbert stratum; Hilbert function
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 \(Hilb^ nR=\cup_{h(1)=n}Z_ h \) [\textit{A. A. Iarrobino}, Mem. Am. Math. Soc. 188 (1977; Zbl 0355.14001), Bull. Am. Math. Soc. 78, 819-823 (1972; Zbl 0268.14002) and \textit{J. Brian??on}, 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 \textit{G. Ellingsrud} and \textit{S. A. Str??mme} [Invent. Math. 87, 343-352 (1987; Zbl 0625.14002)].
A.Papantonopoulou
Zbl 0355.14001; Zbl 0268.14002; Zbl 0353.14004; Zbl 0625.14002