Rank Zero Segre Integrals on Hilbert Schemes of Points on Surfaces

The generating function of the Segre integrals on Hilbert schemes of points on a surface X can be determined by five universal series A₀(z), A₁(z), A₂(z), A₃(z), A₄(z), due to the result of Ellingsrud–Göttsche–Lehn. These five series do not depend on the surface X and depend on the element α ∈ K(X), to which the Segre integrals are associated, only through the rank. Marian–Oprea–Pandharipande have determined A₀(z),A₁(z),A₂(z) for all ranks. For rank 0, it is easy to see A₄(z)=1. Marian–Oprea–Pandharipande also conjectured that A₃(z)=A₀(z)A₁(z) for rank 0. We prove this conjecture by showing that when X is the projective plan, the Segre integrals associated to the structure sheaf of a curve in the anti-canoncial class are all zero. Hence, the rank zero Segre integrals on the Hilbert schemes of points for all surfaces are determined.