Evidence for a conjecture of Ellingsrud and Strømme on the Chowring of Hilbd(P2).Illinois J. Math. 32 (1988), no. 2, 171–210.

The author studies the cohomology ring of HilbdP2, the scheme parametrizing length-d subschemes of P2. Let p: I to HilbdP2 denote the universal family and q: I to P2 the natural projection. For any line bundle L on P2, let E(L) := pqL, a rank d bundle on HilbdP2. Then the conjecture of the title is this: {cj(E(O(m)),m = 0, 1, 2} generate the ring A .(HilbdP2). As “evidence” for this conjecture, the author proves that the monomials of appropriate weights in the Chern classes above generate Aj(HilbdP2), j = 1, 2, 3, d 3. Part 1 of this paper consists of a detailed general study of certain constructions of varieties of nth-order data on a nonsingular surface X. For varieties of first-, second-, and third-order data, two approaches are taken: (1) varieties F, S, and T are defined which are desingularizations of the closures of the open subsets of (respectively) Hilb2X, Hilb3X, Hilb4X that parametrize linear subschemes of X supported at a single point and (2) an iterative construction is made to define bundles F(m) over X. The F(m)’s are modern generalizations of bundles over P2 due to J. G. Semple [Proc. London Math. Soc. (3) 4 (1954), 24–49; MR0061406 (15,820c)]. Both approaches yield isomorphic varieties of first- and second-order data, but give distinct P1-bundles of thirdorder data fibred over second-order data (Proposition 1.6.5, p. 182). As the author remarks, these constructions, particularly those of the varieties F, S, and T, could potentially be quite useful for further applications in algebraic and enumerative geometry. Parts 2 and 3 of the paper are proofs of the author’s main results regarding generation of Aj(HilbdP2), j = 1, 2, 3. The method is a direct (though delicate) one: using F, S, and T, the author finds explicit “candidate” bases for Aj(HilbdP2), j = 1, 2, 3, computes the matrix of intersection degrees of the proposed bases with monomials in the desired Chern classes, and finds minors of the appropriate ranks whose determinants generate the ideal (1) in Z. Then Poincar´e duality can be used...