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The Betti numbers of the Hilbert scheme of points on a smooth projective surface. (English) Zbl 0679.14007

Let S be a smooth projective surface (over \({\mathbb{C}})\) and \(S^{[n]}\) the component of the Hilbert scheme of S which parametrizes subschemes of finite length n on S. Recently G. Ellingsrud and S. A. Strømme have computed the Betti numbers of \(S^{[n]}\) for \(S={\mathbb{P}}_ 2\) [Invent. Math. 87, 343-352 (1987; Zbl 0625.14002)].
In this paper the Betti numbers of \(S^{[n]}\) are computed for an arbitrary smooth projective surface S. For \(X=S\) or \(X=S^{[n]}\) let \(b_ i(X)\) be the i-th Betti number and \(p(X,z)=\sum_{i}b_ i(X)z^ i \) the Poincaré polynomial. Then the main result is: \[ \sum^{\infty}_{n=0}p(S^{[n]},z)t^ n=\prod^{\infty}_{k=1}\frac{(1+z^{2k-1}t^ k)^{b_ 1(S)}(1+z^{2k+1}t^ k)^{b_ 1(S)}}{(1-z^{2k-2}t^ k)^{b_ 0(S)}(1-z^{2k}t^ k)^{b_ 2(S)}(1-z^{2k+2}t^ k)^{b_ 0(S)}}. \] The proof uses reduction modulo q and the Weil conjectures.
We also make some remarks about the Hodge numbers.
Reviewer: L.Göttsche

MSC:

14F45 Topological properties in algebraic geometry
14C05 Parametrization (Chow and Hilbert schemes)

Citations:

Zbl 0625.14002

Software:

goettsche.lib
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References:

[1] Beauville, A.: Variétés kählériennes dont la première classe de Chern est nulle. J. Diff. Geom.18, 755-782 (1983) · Zbl 0537.53056
[2] Bialynicki-Birula, A.: Some theorems on actions of algebraic groups. Ann. Math.98, 480-497 (1973) · Zbl 0275.14007 · doi:10.2307/1970915
[3] Bialynick-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, 667-674 (1976) · Zbl 0355.14015
[4] Briançon, J.: Description de Hilb n C{x, y}. Invent. Math.41, 45-89 (1977) · Zbl 0353.14004 · doi:10.1007/BF01390164
[5] Deligne, P.: la conjecture de Weil. I. Inst. Hautes Etudes Sci. Publ. Math.43, 273-307 (1974) · Zbl 0287.14001 · doi:10.1007/BF02684373
[6] Ellingsrud, G., Strømme, S.A.: On the homology of the Hilbert scheme of points in the plane. Invent. Math.87, 343-352 (1987) · Zbl 0625.14002 · doi:10.1007/BF01389419
[7] Fogarty, J.: Algebraic families on an algebraic surface. Am. J. Math.10, 511-521 (1968) · Zbl 0176.18401 · doi:10.2307/2373541
[8] Fogarty, J.: Algebraic families on an algebraic surface. II. Picard scheme of the punctual Hilbert scheme. Am. J. Math.96, 660-687 (1974) · Zbl 0299.14020
[9] Fulton, W.: Intersection theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. Berlin Heidelberg New York: Springer 1984
[10] Fujiki, A.: On primitive symplectic compact KählerV-manifolds of dimension four. Classification of algebraic and analytic manifolds (Katata 1981). Prog. Math. 59. Boston: Birkhäuser 1983
[11] Göttsche, L.: Die Betti-Zahlen des Hilbert-Schemas für Unterschemata der Längen einer glatten Fläche. Diplomarbeit, Bonn, Juli 1988
[12] Grothendieck, A.: Techniques de construction et théorèmes d’existence en géometrie algébrique. IV. Les schémas de Hilbert. Sém. Bourbaki221 (1960/61)
[13] Hirschowitz, A.: Le groupe de Chow équivariant. C.R. Acad. Sci. Paris Sér. 1298, 87-89 (1984) · Zbl 0563.14001
[14] Iarrobino, A.: Punctual Hilbert schemes. Mem. Am. Math. Soc.188 (1977) · Zbl 0355.14001
[15] Iarrobino, A.: Punctual Hilbert schemes. Bull. Am. Math. Soc.78, 819-823 (1972) · Zbl 0268.14002 · doi:10.1090/S0002-9904-1972-13049-0
[16] Iarrobino, A.: Hilbert scheme of points: overview of last ten years. Proc. of Symp. in Pure Math., Vol. 46, Part 2, Algebraic Geometry, Bowdoin 297-320 (1987) · Zbl 0646.14002
[17] Macdonald, I.G.: The Poincaré polynomial of a symmetric product. Proc. Camb. Phil. Soc.58, 563-568 (1962) · Zbl 0121.39601 · doi:10.1017/S0305004100040573
[18] Mazur, B.: Eigenvalues of Frobenius acting on algebraic varieties over finite fields. Proc. of Symp. in Pure Math., Vol. 29, Algebraic Geometry, Arcata 231-261 (1974)
[19] Milne, J.S.: Étale Cohomology. Princeton Math. Series 33. Princeton: Princeton University Press 1980 · Zbl 0433.14012
[20] Grothendieck, A. et al.: Séminaire de Géometrie Algébrique 1: Revêtements étales et groupe fondamental (1960-61). (Lecture Notes in Math. 224). Berlin Heidelberg New York: Springer 1971
[21] Steenbrink, J.H.M.: Mixed Hodge structures on the vanishing cohomology. Nordic Summer School, Symposium in Mathematics, Oslo 1970, 525-563. Alphen an den Rijn: Sijthoff and Noordhoff 1977
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