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

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.