Two Point Extremal Gromov-witten Invariants of Hilbert Schemes of Points on Surfaces

Abstract. Given an algebraic surface X, the Hilbert scheme X[n] of n-points on X admits a contraction morphism to the n-fold symmetric product X(n) with the extremal ray generated by a class βn of a rational curve. We determine the two point extremal GW-invariants of X[n] with respect to the class dβn for a simply-connected projective surface X and the quantum first Chern class operator of the tautological bundle on X[n]. The methods used are vertex algebraic description of H∗(X[n]), the localization technique applied to X = P2, and a generalization of the reduction theorem of Kiem-J. Li to the case of meromorphic 2-forms.