Publication: The Chow groups of Hilb 4 P 2 and a base for A 2 ,A 3 ,A 2d−2 ,A 2d−3 of Hilb d P
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1986-10-30
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Elsevier
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Abstract
G. Ellingsrud and S. A. Strømme [Invent. Math. 87 (1987), no. 2, 343–352; see the following review] have proved that the Chow group of the Hilbert scheme Hilb d P 2 is free and have computed the ranks of its homogeneous parts A i (Hilb d P 2 ) . In the present note, the author introduces a family of cycles in Hilb d P 2 and conjectures this family to be a basis of the Chow group. In the case d=3 , this follows from a paper by G. Elencwajg and P. Le Barz [C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 12, 635–638; MR0814963 (87c:14006)]. Here the conjecture is proved in case d=4 , and for any d , in the cases i=2,3,2d−3, 2d−2 . The proof consists in calculations of intersection matrices.
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G. ELENCWAJG et P. LE BARZ, Une base de Pic (Hilb1/2 P2), Comptes rendus, 297, série I, 1983, p. 175-178.
G. ELENCWAJG et P. LE BARZ, Détermination de l'anneau de Chow de Hilb 3 P2, Comptes rendus, 301,série I, 1985, p. 635-638.
[3] G. ELLINSGRUD et S. A. STRÇMME, On the homology ofthe Hilbert scheme of points in the plane, Preprint Séries, n° 13, Universitet i Oslo, 1984.
P. LE BARZ, Validité de certaines formules de géométrie énumérative, Comptes rendus, 289, série A, 1979,p. 755-758.
R. MALLAVIBARRENA, Validité de la formule classique des trisécants stationnaires (à paraître).