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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
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References:
[1] [An] Andrews, G. E.: The Theory of Partitions. Encyclopedia of Mathematics and its Applications. Reading, Massachusetts: Addison-Wesley 1976.
[2] [BB1] Białynicki-Birula, A.: Some theorems on actions of algebraic groups. Ann. Math.98 (1973), 480–497 · Zbl 0275.14007 · doi:10.2307/1970915
[3] [BB2] Białynicki-Birula, A.: Some properties of the decompositions of algebraic varieties determined by actions of a torus Bull. Acad. Pol. Sér. Sci. Math. astron. Phys.24 (1976), 667–674. · Zbl 0355.14015
[4] [Br] Briançon, J.: Description deHilb n C{x, y}. Invent. Math.41 (1977), 45–89 · Zbl 0353.14004 · doi:10.1007/BF01390164
[5] [E-S] Ellingsrud, G. and Strømme, S. A.: On the homology of the Hilbert scheme of points in the plane. Invent. Math.87 (1987), 343–352. · Zbl 0625.14002 · doi:10.1007/BF01389419
[6] [F1] Fogarty, J.: Algebraic families on an algebraic surface. Am. J. Math.10 (1968), 511–521 · Zbl 0176.18401 · doi:10.2307/2373541
[7] [F2] Fogarty, J.: Algebraic families on an algebraic surface II: Picard scheme of the punctual Hilbert scheme Am. J. Math.96 (1974) 660–687 · Zbl 0299.14020
[8] [Fu] Fulton, W.: Intersection theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. Berlin-Heidelberg-New York: Springer 1984 · Zbl 0541.14005
[9] [Gr] Grothendieck, A.: Techniques de construction et théorèmes d’existence en géometrie algébrique IV: Les schémas de Hilbert. Sém. Bourbaki221 (1960/61)
[10] [I1] Iarrobino, A.: Punctual Hilbert schemes. Mem. Am. Math. Soc.188 (1977) · Zbl 0355.14001
[11] [I2] Iarrobino, A.: Punctual Hilbert schemes. Bull. Am. Math. Soc.78 (1972), 819–823 · Zbl 0268.14002 · doi:10.1090/S0002-9904-1972-13049-0
[12] [I3] Iarrobino, A.: Hilbert Scheme of Points: Overview of Last Ten Years. Proc. of Symp. in Pure Math. Vol.46 Part 2, Algebraic Geometry, Bowdoin 1987, 297–320 · Zbl 0646.14002
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