×

zbMATH — the first resource for mathematics

On the homology of the Hilbert scheme of points in the plane. (English) Zbl 0625.14002
The authors calculate the Betti numbers of the Hilbert scheme of points in the plane. Observe that the maximal torus of SL(3) acts on \(Hilb^ d({\mathbb{P}}^ 2)\) with isolated fixed points. It follows from a result of Birula-Białynicki that \(Hilb^ d({\mathbb{P}}^ 2)\) has a cellular decomposition. Then the calculation of the Betti numbers reduces to a careful study of the representation of the torus at the tangent spaces of the fixed points. As a by-product to their method, the authors also obtain similar results about the punctual Hilbert scheme and the Hilbert scheme of points in the affine plane.
Reviewer: L.Ein

MSC:
14C05 Parametrization (Chow and Hilbert schemes)
14N05 Projective techniques in algebraic geometry
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] [B1] Bialynicki-Birula, A.: Some theorems on actions of algebraic groups. Ann. Math.98, 480-497 (1973) · Zbl 0275.14007 · doi:10.2307/1970915
[2] [B2] Bialynicki-Birula, A.: Some properties of the decompositions of algebraic varieties determined by actions of a torus. Bull. Acad. Pol. Sci. Sér. Sci. Math. astron. Phys.24 (No. 9) 667-674 (1976) · Zbl 0355.14015
[3] [F1] Fogarty, J.: Algebraic families on an algebraic surface. Am. J. Math.10, 511-521 (1968) · Zbl 0176.18401 · doi:10.2307/2373541
[4] [F2] Fogarty, J.: Algebraic families on an algebraic surface II: Picard scheme of the punctual Hilbert scheme. Am. J. Math.96, 660-687 (1979) · Zbl 0299.14020
[5] [Br] Briancon, J.: Description de Hilb n C{x, y} Invent. Math.41, 45-89 (1977) · Zbl 0353.14004 · doi:10.1007/BF01390164
[6] [Fu] Fulton, W.: Intersection theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. Berlin-Heidelberg-New York: Springer 1984 · Zbl 0541.14005
[7] [Ga] Gaffney, T.: Multiple points and associated ramification loci. In: Singularities. Proceedings of symposia in pure mathematics of the AMS. Volume 40, part 1 (1983)
[8] [Gr] Grothendieck, A.: Techniques de construction et théorèmes d’existence en gèometrie algébrique IV: Les schémas de Hilbert. Sem. Bourbaki221 (1960/61)
[9] [H] Hirschowitz, A.: Le group de Chow équivariant. C.R. Acad. Sc. Paris298, 87 (1984) · Zbl 0563.14001
[10] [I1] Iarrobino, A.: Punctual Hilbert schemes. Mem. Am. Math. Soc.188 (1977) · Zbl 0355.14001
[11] [I2] Iarrobino, A.: Deforming complete intersection Artin algebras. In Singularities, Proceedings of symposia in pure mathematics of the AMS. Volume 40, part 1 (1983) · Zbl 0563.13010
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.