×

zbMATH — the first resource for mathematics

Cellular decomposition of varieties parametrizing homogeneous ideals of \(\mathbb{C}[[x,y]]\). Incidence of cells. I. (Décomposition cellulaire de variétés paramétrant des idéaux homogènes de \(\mathbb{C}[[x,y]]\). Incidence des cellules. I.) (French) Zbl 0812.14003
Let \(G_ T\) be the family of all zero-dimensional homogeneous ideals of \(\mathbb{C} [x,y]\) that have a fixed Hilbert function \(T\). \(G_ T\) is a smooth irreducible algebraic variety and a closed subscheme of the Hilbert scheme parametrizing the zero-dimensional subschemes of the complex plane supported by the origin [A. A. Iarrobino, “Punctual Hilbert schemes”, Mem. Am. Math. Soc. 188 (1977; Zbl 0355.14001)]. There is a cellular decomposition of \(G_ T\) introduced by L. Göttsche [Manuscr. Math. 66, No. 3, 253-259 (1990; Zbl 0714.14004)]. The author recovers such a cellular decomposition using “escaliers”, and studies incidence between cells.

MSC:
14C05 Parametrization (Chow and Hilbert schemes)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14B20 Formal neighborhoods in algebraic geometry
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] Bialynicki-Birula, A. : Some properties of the decompositions of algebraic varieties determined by actions of a torus . Bulletin de l’académie Polonaise des Sciences, Séries des Sciences math. astr. et phys. 24, #9, 667-674 (1976). · Zbl 0355.14015
[2] Briançon, J. : Description de HilbnCx, y . Invent. Math. 41, 45-89 (1977). · Zbl 0353.14004 · doi:10.1007/BF01390164 · eudml:142485
[3] Briançon, J. : Weierstrass préparé à la Hironaka . Astérisque 7, 8, 67-73 (1973). · Zbl 0297.32004
[4] Briançon, J. and Galligo, A. : Déformations distinguées d’un point de C 2 ou R2 . Astérisque 7, 8, 129-138 (1973). · Zbl 0291.14004
[5] Fulton, W. : Intersection theory . Springer-Verlag, New-York, 1984. · Zbl 0541.14005
[6] Griffiths, P. and Harris, J. : Principles of Algebraic Geometry (1977), J. Wiley, New York. · Zbl 0408.14001
[7] Göttsche, L. : Betti-numbers for the Hilbert function strata of the punctual Hilbert scheme in two variables . Manuscripta Math. 66, 253-259 (1990). · Zbl 0714.14004 · doi:10.1007/BF02568495 · eudml:155471
[8] Granger, M. : Géométrie des schémas de Hilbert ponctuels . Mém. Soc. Math. Fr. Nouv. Ser. 7-12 (1982- 1983). · Zbl 0534.14002 · doi:10.24033/msmf.293 · numdam:MSMF_1983_2_8__1_0 · eudml:94832
[9] Iarrobino, A. : Punctual Hilbert scheme . Memoirs of AMS, vol. 10, #188 (1977), AMS, Providence. · Zbl 0355.14001
[10] Yaméogo, J. : Sur l’alignement dans les schémas de Hilbert ponctuels du plan . Math. Ann. 285, 511-525 (1989). · Zbl 0662.14001 · doi:10.1007/BF01455071 · eudml:164615
[11] Yaméogo, J. : Décomposition cellulaire des variétés de larrobino . Incidence des cellules -II - Preprint (Nice 1992).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.