Gromov–Witten Invariants of the Hilbert Scheme of Two Points on a Hirzebruch Surface

Abstract

In Gromov–Witten theory the virtual localization method is used only when the invariant curves are isolated under a torus action. In this paper, we explore a strategy to apply the localization formula to compute the Gromov–Witten invariants by carefully choosing the related cycles to circumvent the continuous families of invariant curves when there are any. For the example of the two-pointed Hilbert scheme of Hirzebruch surface $F_1$, we manage to compute some Gromov–Witten invariants, and then by combining with the associativity law of (small) quantum cohomology ring, we succeed in computing all 1- and 2-pointed Gromov–Witten invariants of genus 0 of the Hilbert scheme with the help of [13].