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Evidence for a conjecture of Ellingsrud and Strømme on the Chow ring of \(Hilb_ d(P^ 2)\). (English) Zbl 0706.14001

Recently G. Ellingsrud and S. A. Strømme [see Invent. Math. 87, 343-352 (1987; Zbl 0625.14002)] have computed the homology groups of \(Hilb_ d({\mathbb{P}}^ 2)\) and propose the following conjecture: Let \(p:\;I\to Hilb_ d({\mathbb{P}}^ 2)\) be the universal family, let q: \(I\to {\mathbb{P}}^ 2\) be the natural projection, let \({\mathcal L}\) be a line bundle on \({\mathbb{P}}^ 2\) then E(\({\mathcal L}):=p_*q^*{\mathcal L}\) is a vector bundle of rank d on \(Hilb_ d({\mathbb{P}}^ 2).\)
Conjecture (Ellingsrud and Strømme). The Chern classes of the bundles (E(\({\mathcal O}(m))\), \(m=0,1,2\), generate the homology ring of \(Hilb_ d({\mathbb{P}}^ 2)\). It is easy to see that the conjecture is true for \(Hilb_ 2({\mathbb{P}}^ 2)\). The conjecture has been verified for \(Hilb_ 3({\mathbb{P}}^ 2)\) by G. Ellingsrud and S. A. Strømme [cf. “On the Hilbert scheme of 3 points in the plane”, Lect. Conf. Rocca di Papa (Rome 1985)]. Here we shall prove:
(1) The monomials of weight 1 in the Chern classes of E(\({\mathcal O}(m))\), \(m=0,1,2\), generate \(A^ 1(Hilb_ d({\mathbb{P}}^ 2))\), \(d\geq 3.\)
(2) The monomials of weight 2 in the Chern classes of E(\({\mathcal O}(m))\), \(m=0,1,2\), generate \(A^ 2(Hilb_ d({\mathbb{P}}^ 2))\), \(d\geq 3.\)
(3) The monomials of weight 3 in the Chern classes of E(\({\mathcal O}(m))\), \(m=0,1,2\), generate \(A^ 3(Hilb_ d({\mathbb{P}}^ 2))\), \(d\geq 3\).

MSC:

14C05 Parametrization (Chow and Hilbert schemes)
14C15 (Equivariant) Chow groups and rings; motives

Citations:

Zbl 0625.14002
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