Universal polynomials for tautological integrals on Hilbert schemes

Abstract

We show that tautological integrals on Hilbert schemes of points can be written in terms of universal polynomials in Chern numbers. The results hold in all dimensions, though they strengthen known results even for surfaces by allowing integrals over arbitrary "geometric" subsets (and their Chern-Schwartz-MacPherson classes).We apply this to enumerative questions, proving a generalised Göttsche Conjecture for all singularity types and in all dimensions. So if L is a sufficiently ample line bundle on a smooth variety X, in a general subsystem P d ⊂ |L| of appropriate dimension the number of hypersurfaces with given singularity types is a polynomial in the Chern numbers of (X, L).When X is a surface, we get similar results for the locus of curves with fixed "BPS spectrum" in the sense of stable pairs theory.1 Clearly, a geometric subset P ⊆ X [n] is constructible and has the property that if Z ∈ P , Z ′ ∈ X [n]

Tautological integrals on curvilinear Hilbert schemes

“…For simplicity let Hilb k 0 (C n ) denote the punctual Hilbert scheme of k points on C n defined as the closed subset of Hilb k (C n ) parametrising subschemes supported at the origin. Following Rennemo [34] we define punctual geometric subsets to be the constructible subsets of the punctual Hilbert scheme containing all 0-dimensional schemes of given isomorphism types.…”

“…Let Hilb k 0 (C n ) be the punctual Hilbert scheme defined as the closed subset of (C n ) [k] = Hilb k (C n ) parametrising subschemes supported at the origin. Following Rennemo [34] we define punctual geometric subsets as constructible subsets Q ⊆ Hilb k 0 (C n ) which are union of isomorphism classes of schemes, that is, if ξ ∈ Q and ξ ′ ∈ Hilb k 0 (C n ) are isomorphic (they have isomorphic coordinate rings) then ξ ′ ∈ Q. Geometric subsets of X [k] of type (Q 1 , . .…”

“…, c rk ) = Z α M are called tautological integrals of F [k] . Rennemo [34] shows that these integrals can be expressed in terms of the Chern numbers of X and F. Theorem 1.1 (Rennemo [34]). Let M r,n denote the set of weighted-degree-n monomials in the Chern classes c 1 (F), .…”

“…The proof of [34] is nonconstructive and based on the fact that an element in the cohomology ring of a Grassmannian is a polynomial in the Chern classes of the universal bundle. Lacking a method of obtaining information about this polynomial, there is no apparent way of turning this proof into an algorithm.…”

Tautological integrals on symmetric products of curves

“…J. V. Rennemo [20] gave a generaliztion of the theorem of G. Ellingsrud, L. Gottsche and M. Lehn: when d = 1 and d = 2 the universal property of polynomials in Chern classes of tautological sheaves and tangent bundles holds; when d > 2, one should consider the universal property of integrals of polynomials only in Chern classes of tautological sheaves over geometric subsets of X [n] .…”