Evidence for a conjecture of Ellingsrud and Strømme on the Chow ring of \(Hilb_ d(P^ 2)\)
Recently \textit{G. Ellingsrud} and \textit{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 \textit{G. Ellingsrud} and \textit{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 \(A^ 3(Hilb_ d({\mathbb{P}}^ 2))\), \(d\geq 3\).
scientific article; zbMATH DE number 4158797